# questions

title: the-curve-y-x-e-x-has-minimum-value-equal-to
slug: the-curve-y-x-e-x-has-minimum-value-equal-to-45028
question: <div class="question">The curve $y=x e^{x}$ has minimum value equal to</div>
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-\frac{1}{\mathrm{e}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{\mathrm{e}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\mathrm{e}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">e</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\frac{1}{\mathrm{e}}$</span> </div>

answer:
<div class="solution">Let $y=x e^{x}$. Differentiate both side w.r.t. 'x'.
<br/> <br/>$$
<br/> <br/>\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{e}^{\mathrm{x}}+\mathrm{xe}^{\mathrm{x}}=\mathrm{e}^{\mathrm{x}}(1+\mathrm{x})
<br/> <br/>$$
<br/> <br/>Put $\frac{\mathrm{dy}}{\mathrm{dx}}=0$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow \mathrm{e}^{\mathrm{x}}(1+\mathrm{x})=0 \\
<br/> <br/>\Rightarrow \mathrm{x}=-1
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Now, $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{\mathrm{x}}(1+\mathrm{x})=\mathrm{e}^{\mathrm{x}}(\mathrm{x}+2)$
<br/> <br/>$$
<br/> <br/>\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right)_{(\mathrm{x}=-1)}=\frac{1}{\mathrm{e}}+0&gt;0
<br/> <br/>$$
<br/> <br/>Hence, $\mathrm{y}=\mathrm{xe}^{\mathrm{x}}$ is minimum function and
<br/> <br/>$$
<br/> <br/>\mathrm{y}_{\min }=-\frac{1}{\mathrm{e}} \text { . }
<br/> <br/>$$</div>


title: the-curve-yex-yx0-has-a-vertical-tangent-at
slug: the-curve-yex-yx0-has-a-vertical-tangent-at-43104
question: <div class="question">The curve \(y-e^{x y}+x=0\) has a vertical tangent at</div>
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\((1,1)\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\((0,1)\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\((1,0)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">no point</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((1,0)\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; y-e^{x y}+x=0 \\<br/>&amp; \therefore \frac{d y}{d x}-e^{x y}\left(y+x \frac{d y}{d x}\right)+1=0 \\<br/>&amp; \text {i.e., } \frac{d y}{d x}-y(x+y)-x(x+y) \frac{d y}{d x}+1=0 \\<br/>&amp; \text {i.e., }[1-x(x+y)] \frac{d y}{d x}=y(x+y)-1<br/>\end{aligned}\)<br/>for the vertical tangents<br/>\(1-x(x+y)=0\) i.e., \(\quad y=\frac{1-x^2}{x}\)<br/>\(\therefore \mathrm{x}=1\) and \(\mathrm{y}=0\)</div>


tags: ['Chemistry', 'Coordination Compounds', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
title:
the-d-electron-configurations-of-cr-2-mn-2-fe-2-and-co-2-are-d-4-d-5-d-6-and-d-7-respectively-which-one-of-the-following-will-exhibit-the-lowest-param-1

slug:
the-d-electron-configurations-of-cr-2-mn-2-fe-2-and-co-2-are-d-4-d-5-d-6-and-d-7-respectively-which-one-of-the-following-will-exhibit-the-lowest-param-1-74077

question:
<div class="question">The $d$ -electron configurations of $\mathrm{Cr}^{2+}, \mathrm{Mn}^{2+}, \mathrm{Fe}^{2+}$ and $\mathrm{Co}^{2+}$ are $d^{4}, d^{5}, d^{6}$ and $d^{7}$ respectively. Which one of the following will exhibit the lowest paramagnetic behaviour?  (Atomic no. $\mathrm{Cr}=24, \mathrm{Mn}=25, \mathrm{Fe}=26, \mathrm{Co}=27)$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{Mn}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/vbd2kuAN3rND1Bi_t98U12zf5nOBeglEF_jqJmrl2Tk.original.fullsize.png"/><br/> <br/> <br/>$\therefore$ Since $\mathrm{Co}^{2+}$ has lowest no. of unpaired electrons hence lowest paramagnetic behaviour is shown by $\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$</div>


tags: ['Physics', 'Dual Nature of Matter', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P2.db
title:
the-debroglie-wavelength-of-a-proton-charge-1-6-10-19-c-m-1-6-10-27-kg-accelerated-through-a-potential-difference-of-1-kv-is

slug:
the-debroglie-wavelength-of-a-proton-charge-1-6-10-19-c-m-1-6-10-27-kg-accelerated-through-a-potential-difference-of-1-kv-is-10161

question:
<div class="question">The de-Broglie wavelength of a proton (charge <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>19</mn></mrow></msup><mo> </mo><mi mathvariant="normal">C</mi><mo>,</mo><mi>m</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>27</mn></mrow></msup><mo> </mo><mi>kg</mi></math>) accelerated through a potential difference of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi>kV</mi></math> is </div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>600</mn><mo> </mo><mo>Å</mo></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>9</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>12</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo> </mo><mo>Å</mo></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>9</mn><mo> </mo><mi>nm</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>9</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>12</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi></math></span> </div>

answer:
<div class="solution"><p>de-Broglie wavelength,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mfrac><mi>h</mi><mi>p</mi></mfrac><mo>=</mo><mfrac><mi>h</mi><msqrt><mn>2</mn><mi>m</mi><mi>E</mi></msqrt></mfrac><mo>=</mo><mfrac><mi>h</mi><msqrt><mn>2</mn><mi>m</mi><mi>q</mi><mi>V</mi></msqrt></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>λ</mi><mo>=</mo><mfrac><mrow><mn>6</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>34</mn></mrow></msup></mrow><msqrt><mn>2</mn><mo>×</mo><mn>1</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>27</mn></mrow></msup><mo>×</mo><mn>1</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>19</mn></mrow></msup><mo>×</mo><mn>1000</mn></msqrt></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>λ</mi><mo>=</mo><mfrac><mrow><mn>6</mn><mo>.</mo><mn>6</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>34</mn></mrow></msup></mrow><mrow><mn>7</mn><mo>.</mo><mn>16</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>22</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>9</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>12</mn></mrow></msup><mo> </mo><mi mathvariant="normal">m</mi></math></p></div>


tags: ['Physics', 'Dual Nature of Matter', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P2.db
title:
the-debroglie-wavelength-of-an-electron-moving-with-a-velocity-3-c-c-3-1-0-8-m-s-is-equal-to-the-wavelength-of-photon-the-ratio-of-the-kinetic-energie

slug:
the-debroglie-wavelength-of-an-electron-moving-with-a-velocity-3-c-c-3-1-0-8-m-s-is-equal-to-the-wavelength-of-photon-the-ratio-of-the-kinetic-energie-53560

question:
<div class="question">The de-Broglie wavelength of an electron moving with a velocity $\frac{c}{3}\left(c=3 \times 10^8 \mathrm{~m} / \mathrm{s}\right)$ is equal to the wavelength of photon. The ratio of the kinetic energies of electron and photon is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$1: 4$
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>$1: 3$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$1: 2$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$2: 1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$1: 3$
<br/></span> </div>

answer:
<div class="solution">The de-Broglie wavelength, $\lambda=\frac{h}{m v}$
<br/>Here, $\quad \lambda_e=\frac{h}{m_e \frac{c}{3}}$ and $\lambda_p=\frac{h}{m_p c}$
<br/>Given, $\quad \lambda_e=\lambda_p$
<br/>So, $\quad \frac{h}{m_e \frac{c}{3}}=\frac{h}{m_p c}$
<br/>$\Rightarrow \quad \frac{m_e}{m_p}=3$
<br/>Ratio of KE $=\frac{K_e}{K_p}=\frac{\frac{1}{2} m_e v_e^2}{\frac{1}{2} m_p v_p^2}=\frac{1}{3}$
<br/>$\left(\because \frac{m_e}{m_p}=3, v_e=\frac{c}{3}\right.$ and $\left.v_p=c\right)$</div>


title: the-decreasing-order-of-the-ionization-potential-of-the-following-elements-is
slug: the-decreasing-order-of-the-ionization-potential-of-the-following-elements-is-91832
tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
question:
<div class="question">The decreasing order of the ionization potential of the following elements is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad \mathrm{Ne}&gt;\mathrm{Cl}&gt;\mathrm{P}&gt;\mathrm{S}&gt;\mathrm{Al}&gt;\mathrm{Mg}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\quad \mathrm{Ne}&gt;\mathrm{Cl}&gt;\mathrm{P}&gt;\mathrm{S}&gt;\mathrm{Mg}&gt;\mathrm{Al}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad \mathrm{Ne}&gt;\mathrm{Cl}&gt;\mathrm{S}&gt;\mathrm{P}&gt;\mathrm{Mg}&gt;\mathrm{Al}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{Ne}&gt;\mathrm{Cl}&gt;\mathrm{S}&gt;\mathrm{P}&gt;\mathrm{Al}&gt;\mathrm{Mg}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad \mathrm{Ne}&gt;\mathrm{Cl}&gt;\mathrm{P}&gt;\mathrm{S}&gt;\mathrm{Mg}&gt;\mathrm{Al}$</span> </div>

answer:
<div class="solution">Closed shell (Ne), half filled (P) and completely filled configuration (Mg) are the cause of higher value of I.E.</div>


tags: ['Chemistry', 'Solutions', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P2.db
title:
the-degree-of-dissociation-of-ca-no-3-2-in-dilute-aq-solution-containing-70-g-of-salt-per-100-g-of-water-at-10-0-c-is-70-if-vapour-pressure-of-water-a

slug:
the-degree-of-dissociation-of-ca-no-3-2-in-dilute-aq-solution-containing-70-g-of-salt-per-100-g-of-water-at-10-0-c-is-70-if-vapour-pressure-of-water-a-52684

question:
<div class="question">The degree of dissociation of $\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2}$ in dilute aq. solution containing $7.0 \mathrm{~g}$ of salt per $100 \mathrm{~g}$ of water at $100^{\circ} \mathrm{C}$ is $70 \%$. If vapour pressure of water at $100^{\circ} \mathrm{C}$ is $760 \mathrm{~mm} \mathrm{Hg}$. The vapour pressure of solution is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">735</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad 730$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">760</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\quad 746$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad 746$</span> </div>

answer:
<div class="solution">$\mathrm{Ca}\left(\mathrm{NO}_{3}\right)_{2} \rightleftharpoons \mathrm{Ca}^{2+}+2 \mathrm{NO}_{3}^{-}$<br/>$1-0.71 \quad 0.7 \quad 2 \times 0.7$<br/>$\quad\left(\because \alpha=\frac{70}{100}=0.7\right)$<br/>$\quad \mathrm{n}_{1}=\frac{100}{18}=5.55$<br/>$\frac{\mathrm{p}^{0}-\mathrm{p}_{\mathrm{s}}}{\mathrm{p}^{0}}=\frac{\mathrm{n}_{2} \times \mathrm{i}}{\mathrm{n}_{1}+\mathrm{n}_{2}}, \frac{760-\mathrm{p}_{\mathrm{s}}}{760}$<br/>$=\frac{0.042 \times 2.4}{5.55+0.042}$<br/>$\therefore \mathrm{p}_{\mathrm{s}}-746 \mathrm{~mm} \mathrm{Hg} .$</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
the-degree-of-dissociation-of-pcl-5-obeying-the-equilibrium-pcl-5-pcl-3-cl-2-is-related-to-the-equilibrium-pressure-by-1

slug:
the-degree-of-dissociation-of-pcl-5-obeying-the-equilibrium-pcl-5-pcl-3-cl-2-is-related-to-the-equilibrium-pressure-by-1-56417

question:
<div class="question">The degree of dissociation of $\mathrm{PCl}_{5}(\alpha)$ obeying
<br/> <br/>the equilibrium $\mathrm{PCl}_{5} \rightleftharpoons \mathrm{PCl}_{3}+\mathrm{Cl}_{2}$ is related to the equilibrium pressure by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\alpha \propto \frac{1}{\mathrm{P}^{4}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\alpha \propto \frac{1}{\sqrt{\mathrm{P}}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\alpha \propto \frac{1}{\mathrm{P}^{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\alpha \propto \mathrm{P}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\alpha \propto \frac{1}{\sqrt{\mathrm{P}}}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/CH0qTfl1P3Qd2qDlQOHZysyowccoOImkNqBE-GV84UA.original.fullsize.png"/><br/> <br/> <br/>$\therefore \mathrm{K}_{\mathrm{p}}=\frac{\frac{\alpha}{1+\alpha} \mathrm{P} \times \frac{\alpha}{1+\alpha} \mathrm{P}}{\frac{1-\alpha}{1+\alpha} \mathrm{P}}=\frac{\alpha^{2} \mathrm{P}}{1-\alpha^{2}}$
<br/> <br/>or, $\mathrm{K}_{\mathrm{p}}=\alpha^{2} \mathrm{P}$
<br/> <br/>$\therefore \alpha=\sqrt{\frac{\mathrm{K}_{\mathrm{p}}}{\mathrm{P}}}$ when $1-\alpha^{2}=1$</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
title:
the-degree-of-dissociation-of-pcl-5-obeying-the-equilibrium-pcl-5-pcl-3-cl-2-is-related-to-the-equilibrium-pressure-by

slug:
the-degree-of-dissociation-of-pcl-5-obeying-the-equilibrium-pcl-5-pcl-3-cl-2-is-related-to-the-equilibrium-pressure-by-29330

question:
<div class="question">The degree of dissociation of $\mathrm{PCl}_{5}(\alpha)$ obeying
<br/> <br/>the equilibrium $\mathrm{PCl}_{5} \rightleftharpoons \mathrm{PCl}_{3}+\mathrm{Cl}_{2}$ is related to the equilibrium pressure by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\alpha \propto \frac{1}{\mathrm{P}^{4}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\alpha \propto \frac{1}{\sqrt{\mathrm{P}}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\alpha \propto \frac{1}{\mathrm{P}^{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\alpha \propto \mathrm{P}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\alpha \propto \frac{1}{\sqrt{\mathrm{P}}}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/CH0qTfl1P3Qd2qDlQOHZysyowccoOImkNqBE-GV84UA.original.fullsize.png"/><br/> <br/> <br/>$\therefore \mathrm{K}_{\mathrm{p}}=\frac{\frac{\alpha}{1+\alpha} \mathrm{P} \times \frac{\alpha}{1+\alpha} \mathrm{P}}{\frac{1-\alpha}{1+\alpha} \mathrm{P}}=\frac{\alpha^{2} \mathrm{P}}{1-\alpha^{2}}$
<br/> <br/>or, $\mathrm{K}_{\mathrm{p}}=\alpha^{2} \mathrm{P}$
<br/> <br/>$\therefore \alpha=\sqrt{\frac{\mathrm{K}_{\mathrm{p}}}{\mathrm{P}}}$ when $1-\alpha^{2}=1$</div>


title: the-degree-of-the-differential-equation-leftfracd3-yd-x3right2-343-fracd2-yd-x25-fracd-yd-x0-is
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P2.db
slug:
the-degree-of-the-differential-equation-leftfracd3-yd-x3right2-343-fracd2-yd-x25-fracd-yd-x0-is-57918

question:
<div class="question">The degree of the differential equation \(\left(\frac{d^3 y}{d x^3}\right)^{2 / 3}+4-3 \frac{d^2 y}{d x^2}+5 \frac{d y}{d x}=0\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \left(\frac{d^3 y}{d x^3}\right)^{2 / 3}+4-3 \frac{d^2 y}{d x^2}+5 \frac{d y}{d x}=0 \\<br/>&amp; \Rightarrow\left(\frac{d^3 y}{d x^3}\right)^2=\left[3 \frac{d^2 y}{d x^2}-5 \frac{d y}{d x}-4\right]^3<br/>\end{aligned}\)<br/>It is a differential equation of degree 2.</div>


title: the-degree-of-the-differential-equation-satisfying-1-x-2-1-y-2-a-x-y-is
slug: the-degree-of-the-differential-equation-satisfying-1-x-2-1-y-2-a-x-y-is-84468
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The degree of the differential equation satisfying
<br/> <br/>$\sqrt{1-x^{2}}+\sqrt{1+y^{2}}=a(x-y)$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">$\quad$ Put $x=\sin \theta$ and $y=\sin \phi$
<br/> <br/>$\Rightarrow \cos \theta+\cos \phi=a(\sin \theta-\sin \phi)$
<br/> <br/>$\Rightarrow 2 \cos$
<br/> <br/>$\begin{aligned} &amp; \frac{\theta+\phi}{2} \cos \frac{\theta-\phi}{2}=2 a \cos \frac{\theta+\phi}{2} \sin \frac{\theta-\phi}{2} \\ \Rightarrow \cot \frac{\theta-\phi}{2}=a \Rightarrow \theta-\phi=2 \cot ^{-1} \mathrm{a} \\ \Rightarrow \sin ^{-1} \mathrm{x}-\sin ^{-1} \mathrm{y}=2 \cot ^{-1} \mathrm{a} \\ \text { Differentiate } \frac{1}{\sqrt{1-\mathrm{x}^{2}}}-\frac{1}{\sqrt{1-\mathrm{y}^{2}}} \frac{\mathrm{dy}}{\mathrm{dx}}=0 \end{aligned}$
<br/> <br/>so the degree is one</div>


tags: ['Chemistry', 'States of Matter', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
title:
the-density-of-a-certain-gas-at-30-0-c-and-768-torr-is-1-35-kg-m-3-then-density-of-the-gas-at-stp-is

slug:
the-density-of-a-certain-gas-at-30-0-c-and-768-torr-is-1-35-kg-m-3-then-density-of-the-gas-at-stp-is-90649

question:
<div class="question">The density of a certain gas at <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>30</mn><mn>0</mn></msup><mi mathvariant="normal">C</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>768</mn></math> torr is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>35</mn><mo> </mo><mi>kg</mi><mo>/</mo><msup><mi mathvariant="normal">m</mi><mn>3</mn></msup></math>, then density of the gas at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>STP</mi></math> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>48</mn><mo> </mo><mi>kg</mi><mo>/</mo><msup><mi mathvariant="normal">m</mi><mn>3</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>27</mn><mo> </mo><mi>kg</mi><mo>/</mo><msup><mi mathvariant="normal">m</mi><mn>3</mn></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>35</mn><mo> </mo><mi>kg</mi><mo>/</mo><msup><mi mathvariant="normal">m</mi><mn>3</mn></msup></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>00</mn><mo> </mo><mi>kg</mi><mo>/</mo><msup><mi mathvariant="normal">m</mi><mn>3</mn></msup></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>48</mn><mo> </mo><mi>kg</mi><mo>/</mo><msup><mi mathvariant="normal">m</mi><mn>3</mn></msup></math></span> </div>

answer:
<div class="solution"><p>Using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>PM</mi><mo>=</mo><mi>dRT</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi mathvariant="normal">P</mi><mn>1</mn></msub><mi mathvariant="normal">M</mi></mrow><mrow><msub><mi mathvariant="normal">P</mi><mn>2</mn></msub><mi mathvariant="normal">M</mi></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><msub><mi mathvariant="normal">d</mi><mn>1</mn></msub><msub><mi>RT</mi><mn>1</mn></msub></mrow><mrow><msub><mi mathvariant="normal">d</mi><mn>2</mn></msub><msub><mi>RT</mi><mn>2</mn></msub></mrow></mfrac><mspace linebreak="newline"></mspace><mfrac><mn>768</mn><mn>760</mn></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mn>1</mn><mo>.</mo><mn>35</mn><mo>×</mo><mn>303</mn></mrow><mrow><msub><mi mathvariant="normal">d</mi><mn>2</mn></msub><mo>×</mo><mn>273</mn></mrow></mfrac><mspace linebreak="newline"></mspace><msub><mi mathvariant="normal">d</mi><mn>2</mn></msub><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mn>1</mn><mo>.</mo><mn>35</mn><mo>×</mo><mn>303</mn><mo>×</mo><mn>760</mn></mrow><mrow><mn>273</mn><mo>×</mo><mn>768</mn></mrow></mfrac><mspace linebreak="newline"></mspace><msub><mi mathvariant="normal">d</mi><mn>2</mn></msub><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo>.</mo><mn>48</mn><mo> </mo><mi>kg</mi><mo>/</mo><msup><mi mathvariant="normal">m</mi><mn>3</mn></msup><mspace linebreak="newline"></mspace></math></p></div>


title: the-derivative-of-e-x-3-with-respect-to-lo-g-x-is
slug: the-derivative-of-e-x-3-with-respect-to-lo-g-x-is-21659
question: <div class="question">The derivative of $e^{x^3}$ with respect to $\log x$ is</div>
tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{e}^{\mathrm{x}^3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \mathrm{x}^2 2 \mathrm{e}^{\mathrm{x}^3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$3 \mathrm{x}^3 \mathrm{e}^{\mathrm{x}^3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 \mathrm{x}^3 \mathrm{e}^{\mathrm{x}^3}+3 \mathrm{x}^2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 \mathrm{x}^3 \mathrm{e}^{\mathrm{x}^3}$</span> </div>

answer:
<div class="solution">Let $y=\mathrm{e}^{\mathrm{x}^3}, \mathrm{z}=\log \mathrm{x}$<br/>On differentiating w.r.t.x, we get<br/>$\frac{d y}{d x}=e^{x^3}\left(3 x^2\right)=3 x^2 e^{x^3} \text { and } \frac{d z}{d x}=\frac{1}{x}$<br/>$\therefore \frac{d y}{d z}=\frac{\frac{d y}{d x}}{\frac{d z}{d x}}=\frac{3 x^2 e^{x^3}}{\left(\frac{1}{x}\right)}=3 x^3 e^{x^3}$</div>


title: the-determinant-1-1-1-x-3-x-4-x-5-x-3-2-x-4-2-x-5-2
slug: the-determinant-1-1-1-x-3-x-4-x-5-x-3-2-x-4-2-x-5-2-61507
tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P2.db
question:
<div class="question">The determinant $\left|\begin{array}{ccc}1 &amp; (x-3) &amp; (x-3)^{2} \\ 1 &amp; (x-4) &amp; (x-4)^{2} \\ 1 &amp; (x-5) &amp; (x-5)^{2}\end{array}\right|$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3 values of $x$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 values of $x$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1 values of $x$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">No value of $x$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">No value of $x$</span> </div>

answer:
<div class="solution">The given determinant vanishes, i.e.,
<br/> <br/>$$
<br/> <br/>\left|\begin{array}{ccc}
<br/> <br/>1 &amp; x-3 &amp; (x-3)^{2} \\
<br/> <br/>1 &amp; x-4 &amp; (x-4)^{2} \\
<br/> <br/>1 &amp; x-5 &amp; (x-5)^{2}
<br/> <br/>\end{array}\right|=0
<br/> <br/>$$
<br/> <br/>Expanding along $\mathrm{C}_{1},$ we get
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>(x-4)(x-5)^{2}-(x-5)(x-4)^{2}-\left\{(x-3)(x-5)^{2}\right. \\
<br/> <br/>\left.-(x-5)(x-3)^{2}\right\}+(x-3)(x-4)^{2} \\
<br/> <br/>-(x-4)(x-3)^{2}=0 \\
<br/> <br/>\Rightarrow(x-4)(x-5)(x-5-x+4) \\
<br/> <br/>-(x-3)(x-5)(x-5-x+3) \\
<br/> <br/>+(x-3)(x-4)(x-4-x+3)=0
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>$$
<br/> <br/>\Rightarrow-(x-4)(x-5)+2(x-3)(x-5)-(x-3)
<br/> <br/>$$
<br/> <br/>$(x-4)=0$
<br/> <br/>$$
<br/> <br/>\Rightarrow-x^{2}+9 x-20+2 x^{2}-16 x+30-x^{2}+7 x-12=0
<br/> <br/>$$
<br/> <br/>$\Rightarrow-32+30=0 \Rightarrow-2=0$
<br/> <br/>Which is not possible, hence no value of $\mathrm{x}$ satisfies the given condition.</div>


tags: ['Mathematics', 'Differentiation', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
title:
the-diagonal-of-a-square-is-changing-at-the-rate-of-05-cm-sec-then-the-rate-of-change-of-area-when-the-area-is-400-cm-2-is-equal-to

slug:
the-diagonal-of-a-square-is-changing-at-the-rate-of-05-cm-sec-then-the-rate-of-change-of-area-when-the-area-is-400-cm-2-is-equal-to-56779

question:
<div class="question">The diagonal of a square is changing at the rate of $0.5 \mathrm{~cm} / \mathrm{sec}$. Then the rate of change of area, when the area is $400 \mathrm{~cm}^{2}$, is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$20 \sqrt{2} \mathrm{~cm}^{2} / \mathrm{sec}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$10 \sqrt{2} \mathrm{~cm}^{2} / \mathrm{sec}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{10 \sqrt{2}} \mathrm{~cm}^{2} / \mathrm{sec}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{10}{\sqrt{2}} \mathrm{~cm}^{2} / \mathrm{sec}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$10 \sqrt{2} \mathrm{~cm}^{2} / \mathrm{sec}$</span> </div>

answer:
<div class="solution">Diagonal $\mathrm{D}=\sqrt{2}$ a
<br/>Differentiating w.r.t.
<br/>t: $\frac{\mathrm{dD}}{\mathrm{dt}}=\sqrt{2} \frac{\mathrm{da}}{\text { at }}$
<br/>or $\frac{\mathrm{da}}{\mathrm{dt}}=\frac{1}{\sqrt{2}} \frac{\mathrm{da}}{\mathrm{dt}}=\frac{1}{\sqrt{2}} \times 0.5 \mathrm{~cm} / \mathrm{s}$
<br/>Let Area is denoted by $\underline{A}$ $\frac{\mathrm{d} \mathrm{A}}{\mathrm{dt}}=2 \mathrm{a} \frac{\mathrm{da}}{\mathrm{dt}}$
<br/>when area $A$ is $400 \mathrm{~cm}^{2}$ then $\mathrm{a}=20$ $\therefore \frac{\mathrm{d} \mathrm{A}}{\mathrm{dt}}=2 \times 20 \times \frac{0.5}{\sqrt{2}}=10 \sqrt{2} \mathrm{~cm}^{2} / \mathrm{sec}$</div>


tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2010']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P2.db
title:
the-diameter-of-the-objective-of-a-telescope-is-a-its-magnifying-power-is-m-and-wavelength-of-light-is-the-resolving-power-of-the-telescope-is

slug:
the-diameter-of-the-objective-of-a-telescope-is-a-its-magnifying-power-is-m-and-wavelength-of-light-is-the-resolving-power-of-the-telescope-is-56287

question:
<div class="question">The diameter of the objective of a telescope is a, its magnifying power is $\mathrm{m}$ and wavelength of light is $\lambda,$ The resolving power of the telescope is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(1.22 \lambda) / \mathrm{a}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(1.22 \mathrm{a}) / \lambda$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\lambda \mathrm{m} /(1.22 \mathrm{a})$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">a/(1.22\lambda)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a/(1.22\lambda)</span> </div>


tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
title:
the-difference-in-the-number-of-unpaired-electrons-of-a-metal-ion-in-its-highspin-and-lowspin-octahedral-complexes-is-two-the-metal-ion-is-1

slug:
the-difference-in-the-number-of-unpaired-electrons-of-a-metal-ion-in-its-highspin-and-lowspin-octahedral-complexes-is-two-the-metal-ion-is-1-70688

question:
<div class="question">The difference in the number of unpaired electrons of a metal ion in its high-spin and low-spin octahedral complexes is two. The metal ion is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{Ni}^{2+}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{Fe}^{2+}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{Co}^{2+}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{Mn}^{2+}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Co}^{2+}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/t2h2RYnJ4ecess0kEAG69blc_UnIp8e8LsrZen8I-U4.original.fullsize.png"/></div>


title: the-difference-of-maximum-and-minimum-values-of-f-x-x-2-e-x-is
slug: the-difference-of-maximum-and-minimum-values-of-f-x-x-2-e-x-is-26886
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
question:
<div class="question">The difference of maximum and minimum values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>/</mo><mi>e</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>/</mo><mi>e</mi></math></span> </div>

answer:
<div class="solution"><p>We have, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><msup><mi>e</mi><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><msup><mi>e</mi><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mrow><mo>-</mo><mi>h</mi></mrow></mfenced><mo>&lt;</mo><mn>0</mn><mo>,</mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>h</mi></mfenced><mo>&gt;</mo><mn>0</mn></math>, So minimum at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>h</mi></mrow></mfenced><mo>&gt;</mo><mn>0</mn><mo>,</mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>h</mi></mrow></mfenced><mo>&lt;</mo><mn>0</mn></math>, So maximum at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mrow><mo>-</mo><mn>1</mn><mo>-</mo><mi>h</mi></mrow></mfenced><mo>&gt;</mo><mn>0</mn><mo>,</mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mrow><mo>-</mo><mn>1</mn><mo>+</mo><mi>h</mi></mrow></mfenced><mo>&lt;</mo><mn>0</mn></math>, So maximum at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn></math></p><p>Minimum <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math></p><p>Maximum <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>e</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mi>e</mi></mfrac></math></p><p>Since, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>≥</mo><mn>0</mn><mo>∀</mo><mi>x</mi></math></p><p>So, difference between maximum and minimum values</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mi>e</mi></mfrac><mo>-</mo><mn>0</mn><mo>=</mo><mfrac><mn>1</mn><mi>e</mi></mfrac></math></p></div>


tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
title:
the-differential-equation-whose-solution-is-a-x-2-b-y-2-1-where-a-and-b-are-arbitrary-constants-is-of-1

slug:
the-differential-equation-whose-solution-is-a-x-2-b-y-2-1-where-a-and-b-are-arbitrary-constants-is-of-1-21400

question:
<div class="question">The differential equation whose solution is $A x^{2}+B y^{2}=1$ where $A$ and $B$ are arbitrary constants is of</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">second order and second degree</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">first order and second degree</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">first order and first degree</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">second order and first degree</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">second order and first degree</span> </div>

answer:
<div class="solution">$A x^{2}+B y^{2}=1$ ...(1)
<br/>$A x+B y \frac{d y}{d x}=0$ ...(2)
<br/>$A+B y \frac{d^{2} y}{d x^{2}}+B\left(\frac{d y}{d x}\right)^{2}=0$ ...(3)
<br/>$x\left\{-B y \frac{d^{2} y}{d x^{2}}-B\left(\frac{d y}{d x}\right)^{2}\right\}+B y \frac{d y}{d x}=0$
<br/>Dividing both sides by $-\mathrm{B}$, we get $x y \frac{d^{2} y}{d x^{2}}+x\left(\frac{d y}{d x}\right)^{2}-y \frac{d y}{d x}=0$
<br/>Which is a DE of order 2 and degree 1</div>


tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">72</span> </div>
source_db: MarksBatch2_P2.db
title:
the-digits-mathrma-mathrmb-and-mathrmc-are-such-that-the-three-digit-numbers-a-886-mathrmb-886-mathrmc-are-divisible-by-72-then-the-determinant-leftbe

slug:
the-digits-mathrma-mathrmb-and-mathrmc-are-such-that-the-three-digit-numbers-a-886-mathrmb-886-mathrmc-are-divisible-by-72-then-the-determinant-leftbe-73931

question:
<div class="question">The digits \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) are such that the three digit numbers \(A 88,6 \mathrm{~B} 8,86 \mathrm{C}\) are divisible by 72 then the determinant \(\left|\begin{array}{ccc}\text { A } &amp; 6 &amp; 8 \\ 8 &amp; B &amp; 6 \\ 8 &amp; 8 &amp; C\end{array}\right|\) is divisible by</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">72</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">144</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">288</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">216</span> </li> </ul>

answer:
<div class="solution">\(R_3 \rightarrow 100 R_1+10 R_2+R_3\)<br/>\(\Rightarrow\left|\begin{array}{ccc}<br/>A &amp; 6 &amp; 8 \\<br/>8 &amp; B &amp; 6 \\<br/>8 &amp; 8 &amp; C<br/>\end{array}\right|=\left|\begin{array}{ccc}<br/>A &amp; 6 &amp; 8 \\<br/>8 &amp; B &amp; 6 \\<br/>A 88 &amp; 6 B C &amp; 86 C<br/>\end{array}\right|\)<br/>which is divisible by 72.</div>


title: the-dimensional-formula-of-0-0-is-1
slug: the-dimensional-formula-of-0-0-is-1-28555
tags: ['Physics', 'Units and Dimensions', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
question:
<div class="question">The dimensional formula of <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><msub><mi>μ</mi><mn>0</mn></msub><msub><mi>ε</mi><mn>0</mn></msub></mfrac></msqrt></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi>ML</mi><mn>2</mn></msup><mo> </mo><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup><msup><mi mathvariant="normal">A</mi><mn>2</mn></msup></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">M</mi><mn>0</mn></msup><msup><mi>LT</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">A</mi><mn>0</mn></msup></mrow></mfenced></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi>ML</mi><mn>2</mn></msup><mo> </mo><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup><msup><mi mathvariant="normal">A</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi mathvariant="normal">M</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo> </mo><msup><mi mathvariant="normal">L</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo> </mo><msup><mi mathvariant="normal">T</mi><mn>3</mn></msup><mo> </mo><msup><mi mathvariant="normal">A</mi><mn>2</mn></msup></mrow></mfenced></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><msup><mi>ML</mi><mn>2</mn></msup><mo> </mo><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup><msup><mi mathvariant="normal">A</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p>We know that, velocity of light <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> is terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϵ</mi><mn>0</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mn>0</mn></msub></math> is given by</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mfrac><mn>1</mn><msqrt><msub><mi>μ</mi><mn>0</mn></msub><msub><mi>ε</mi><mn>0</mn></msub></msqrt></mfrac><mspace linebreak="newline"></mspace><mo>⇒</mo><msup><mi>c</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>μ</mi><mn>0</mn></msub><msub><mi>ε</mi><mn>0</mn></msub></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>c</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><msubsup><mi>μ</mi><mn>0</mn><mn>2</mn></msubsup></mfrac><mfrac><msub><mi>μ</mi><mn>0</mn></msub><msub><mi>ε</mi><mn>0</mn></msub></mfrac><mspace linebreak="newline"></mspace><mo>⇒</mo><mi>c</mi><mo>=</mo><mfrac><mn>1</mn><msub><mi>μ</mi><mn>0</mn></msub></mfrac><msqrt><mfrac><msub><mi>μ</mi><mn>0</mn></msub><msub><mi>ε</mi><mn>0</mn></msub></mfrac></msqrt><mspace linebreak="newline"></mspace><mo>⇒</mo><msqrt><mfrac><msub><mi>μ</mi><mn>0</mn></msub><msub><mi>ε</mi><mn>0</mn></msub></mfrac></msqrt><mo>=</mo><mi>c</mi><msub><mi>μ</mi><mn>0</mn></msub><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>Dimension of the magnetic field, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mfrac><mi>τ</mi><mrow><mi>N</mi><mi>I</mi><mi>A</mi></mrow></mfrac><mo>=</mo><mfrac><mfenced close="]" open="["><mrow><msup><mi>ML</mi><mrow><mn>2</mn><mo> </mo></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfenced close="]" open="["><mi>A</mi></mfenced><mfenced close="]" open="["><msup><mi mathvariant="normal">L</mi><mn>2</mn></msup></mfenced></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced close="]" open="["><mrow><msup><mi>MA</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math></p><p>Again, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><msub><mi>μ</mi><mn>0</mn></msub><mi>n</mi><mi>I</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msub><mi>μ</mi><mn>0</mn></msub><mo>=</mo><mfrac><mi>B</mi><mrow><mi>n</mi><mi>l</mi></mrow></mfrac><mo>=</mo><mfrac><mfenced close="]" open="["><mrow><msup><mi>MA</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mrow><mfenced close="]" open="["><msup><mi mathvariant="normal">L</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mfenced close="]" open="["><mi>A</mi></mfenced></mrow></mfrac><mo>=</mo><mfenced close="]" open="["><mrow><msup><mi>MLA</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mn>0</mn></msub><mo>=</mo><mfenced close="]" open="["><mrow><msup><mi>MLA</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>ii</mi></mfenced></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mfenced close="]" open="["><msup><mi>LT</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>iii</mi></mfenced></math></p><p>From Equations. (i), (ii) and (iii), we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><msub><mi>μ</mi><mn>0</mn></msub><msub><mi>ϵ</mi><mn>0</mn></msub></mfrac></msqrt><mo>=</mo><mfenced close="]" open="["><msup><mi>LT</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced><mfenced close="]" open="["><mrow><msup><mi>MLA</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mo>=</mo><mfenced close="]" open="["><mrow><msup><mi>ML</mi><mn>2</mn></msup><msup><mi mathvariant="normal">T</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup><msup><mi mathvariant="normal">A</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced></math></p></div>


title: the-disease-caused-the-high-concentration-of-hydrocarbon-pollutants-in-atmosphere-isare
slug: the-disease-caused-the-high-concentration-of-hydrocarbon-pollutants-in-atmosphere-isare-97602
tags: ['Chemistry', 'Environmental Chemistry', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">cancer</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The disease caused the high concentration of hydrocarbon pollutants in atmosphere is/are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">silicosis</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{TB}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">cancer</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">asthma</span> </li> </ul>

answer:
<div class="solution">The high concentration of hydrocarbon pollutants in atmosphere causes cancer.</div>


title: the-displacement-of-a-particle-is-given-at-time-t-by-x-a-sin-2-t-b-sin-2-t-then
slug: the-displacement-of-a-particle-is-given-at-time-t-by-x-a-sin-2-t-b-sin-2-t-then-25957
tags: ['Physics', 'Oscillations', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P2.db
question:
<div class="question">The displacement of a particle is given at time t, by:
<br/> <br/>$x=A \sin (-2 \omega t)+B \sin ^{2} \omega t \quad$ Then,</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">the motion of the particle is SHM with an amplitude of $\sqrt{A^{2}+\frac{B^{2}}{4}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">the motion of the particle is not SHM, but oscillatory with a time period of $T=\pi / \omega$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">the motion of the particle is oscillatory with a time period of $T=\pi / 2 \omega$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">the motion of the particle is a periodic.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">the motion of the particle is SHM with an amplitude of $\sqrt{A^{2}+\frac{B^{2}}{4}}$</span> </div>

answer:
<div class="solution">The displacement of the particle is given by:
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>x &amp;=A \sin (-2 \omega t)+B \sin ^{2} \omega t \\
<br/> <br/>&amp;=-A \sin 2 \omega t+\frac{B}{2}(1-\cos 2 \omega t) \\
<br/> <br/>&amp;=-\left(A \sin 2 \omega t+\frac{B}{2} \cos 2 \omega t\right)+\frac{B}{2}
<br/> <br/>\end{aligned}
<br/> <br/>$$
<br/> <br/>This motion represents SHM with an amplitude: $\sqrt{A^{2}+\frac{B^{2}}{4}},$ and mean position
<br/> <br/>$$
<br/> <br/>\frac{B}{2}
<br/> <br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
title:
the-displacement-y-of-a-particle-in-a-medium-can-be-expressed-as-y-1-0-6-sin-100-t-20-x-4-m-where-t-is-in-second-and-x-in-meter-the-speed-of-the-wave-

slug:
the-displacement-y-of-a-particle-in-a-medium-can-be-expressed-as-y-1-0-6-sin-100-t-20-x-4-m-where-t-is-in-second-and-x-in-meter-the-speed-of-the-wave-42231

question:
<div class="question">The displacement y of a particle in a medium can be expressed as, $y=10^{-6} \sin \left(100 t+20 x+\frac{\pi}{4}\right) m$ where $t$ is in second and $x$ in meter. The speed of the wave is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$20 \mathrm{~m} / \mathrm{s}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$5 \mathrm{~m} / \mathrm{s}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2000 \mathrm{~m} / \mathrm{s}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5 \pi \mathrm{m} / \mathrm{s}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5 \mathrm{~m} / \mathrm{s}$</span> </div>

answer:
<div class="solution">From equation given,
<br/> <br/>$\begin{array}{l}
<br/> <br/>\omega=100 \\
<br/> <br/>\text { and } \mathrm{k}=20 \\
<br/> <br/>\mathrm{v}=\frac{\omega}{\mathrm{k}}=\frac{100}{20}=5 \mathrm{~m} / \mathrm{s}
<br/> <br/>\end{array}$</div>


title: the-distance-between-the-origin-and-the-normal-to-curve-y-e-2-x-x-2-at-x-0-is
slug: the-distance-between-the-origin-and-the-normal-to-curve-y-e-2-x-x-2-at-x-0-is-30853
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
question:
<div class="question">The distance between the origin and the normal to curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>e</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><msqrt><mn>3</mn></msqrt></mfrac></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><msqrt><mn>5</mn></msqrt></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msqrt><mn>5</mn></msqrt></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><msqrt><mn>5</mn></msqrt></mfrac></math></span> </div>

answer:
<div class="solution"><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>e</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></math></p><p>at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>2</mn><msup><mi>e</mi><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>+</mo><mn>2</mn><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac></mfenced><mrow><mi>x</mi><mo>=</mo><mn>0</mn></mrow></msub><mo>=</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Equation of normal at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math> is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>1</mn><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>(</mo><mi>x</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>-</mo><mn>2</mn><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Distance from <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math> from normal <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>2</mn><msqrt><mn>5</mn></msqrt></mfrac></math></p></div>


title: the-distance-between-the-parallel-lines-3-x-4-y-7-0-and-3-x-4-y-5-0-is-b-a-value-of-a-b-is
slug: the-distance-between-the-parallel-lines-3-x-4-y-7-0-and-3-x-4-y-5-0-is-b-a-value-of-a-b-is-54240
tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">7</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The distance between the parallel lines $3 x-4 y+7=0$ and $3 x-4 y+5=0$ is $\frac{a}{b}$. Value of $a+b$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">5</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">7</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

answer:
<div class="solution">Given parallel lines are
<br/> <br/>$3 x-4 y+7=0$ and $3 x-4 y+5=0$
<br/> <br/>Required distance $=\frac{|7-5|}{\sqrt{(3)^{2}+(-4)^{2}}}=\frac{2}{5}$ $\Rightarrow a=2, b=5$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P2.db
title:
the-distance-from-the-point-345-to-the-point-where-the-line-1-x-3-2-y-4-2-z-5-meets-the-plane-x-y-z-17-is

slug:
the-distance-from-the-point-345-to-the-point-where-the-line-1-x-3-2-y-4-2-z-5-meets-the-plane-x-y-z-17-is-22234

question:
<div class="question">The distance from the point (3,4,5) to the point where the line $\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}$ meets the plane $x+y+z=17$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{2}$</span> </li> </ul>

answer:
<div class="solution">Anypoint on the line is $(r+3,2 r+4,2 r+5)$. It lies on the plane $x+y+z=17$ $\therefore(r+3)+(2 r+4)+(2 r+5)=17$ i.e $r=1$
<br/> <br/>Thus the point of intersection of the plane and the line is (4,6,7) Required distance $=$ distance between (3,4,5) and (4,6,7)
<br/> <br/>$=\sqrt{\left\{(4-3)^{2}+(6-4)^{2}+(7-5)^{2}\right\}}=3$</div>


title: the-distance-of-the-point-11-from-the-line-12x65y2-is
slug: the-distance-of-the-point-11-from-the-line-12x65y2-is-24734
tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The distance of the point \((-1,1)\) from the line \(12(x+6)=5(y-2)\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">The given line is \(12(x+6)=5(y-2)\)<br/>\(\Rightarrow 12 x+72=5 y-10\)<br/>or \(12 x-5 y+72+10=0 \Rightarrow 12 x-5 y+82=0\) The perpendicular distance from \(\left(x_1, y_1\right)\) to the line \(a x+b y+c=0\) is \(\frac{\left(\mathrm{ax}_1+\mathrm{by}_1+\mathrm{c}\right)}{\sqrt{\mathrm{a}^2+\mathrm{b}^2}}\). The point \(\left(x_1, y_1\right)\) is \((-1,1)\), therefore, perpendicular distance from \((-1,1)\) to the line \(12 x-5 y+82=0\) is<br/>\(=\frac{|-12-5+82|}{\sqrt{12^2+(-5)^2}}=\frac{65}{\sqrt{144+25}}=\frac{65}{\sqrt{169}}=5\)</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P2.db
title:
the-distances-of-the-point-1-5-9-from-the-plane-x-y-z-5-measured-along-a-straight-line-x-y-z-is-2-3-k-then-the-value-of-k-is

slug:
the-distances-of-the-point-1-5-9-from-the-plane-x-y-z-5-measured-along-a-straight-line-x-y-z-is-2-3-k-then-the-value-of-k-is-27692

question:
<div class="question">The distances of the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>9</mn><mo>)</mo></math> from the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>5</mn></math> measured along a straight line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>y</mi><mo>=</mo><mi>z</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mn>3</mn></msqrt><mi>k</mi></math>, then the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>3</mn></msqrt></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span> </li> </ul>

answer:
<div class="solution"><p>Given equation of plane is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>5</mn></math>.</p><p>The distance measured along the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>y</mi><mo>=</mo><mi>z</mi></math>.</p><p>Direction ratios of the given line is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></math>.</p><p>So, the equation of line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi></math> is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mn>1</mn></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo>+</mo><mn>5</mn></mrow><mn>1</mn></mfrac><mo>=</mo><mfrac><mrow><mi>z</mi><mo>-</mo><mn>9</mn></mrow><mn>1</mn></mfrac><mo>=</mo><mi>λ</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>x</mi><mo>=</mo><mi>λ</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>=</mo><mi>λ</mi><mo>-</mo><mn>5</mn><mo>,</mo><mi>z</mi><mo>=</mo><mi>λ</mi><mo>+</mo><mn>9</mn></math></p><p>lies on the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>5</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo>(</mo><mi>λ</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>-</mo><mo>(</mo><mo>-</mo><mi>λ</mi><mo>+</mo><mn>5</mn><mo>)</mo><mo>+</mo><mo>(</mo><mi>λ</mi><mo>+</mo><mn>9</mn><mo>)</mo><mo>=</mo><mn>5</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>λ</mi><mo>=</mo><mo>-</mo><mn>10</mn></math></p><p>The coordinate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mo>-</mo><mn>15</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>)</mo></math> and the coordinate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo>-</mo><mn>5</mn><mo>,</mo><mn>9</mn><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi><mo>=</mo><msqrt><mo>(</mo><mn>10</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>10</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>10</mn><msup><mo>)</mo><mn>2</mn></msup></msqrt><mo>=</mo><mn>10</mn><msqrt><mn>3</mn></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mi>k</mi><mo>=</mo><mn>10</mn><msqrt><mn>3</mn></msqrt><mo>⇒</mo><mi>k</mi><mo>=</mo><mn>5</mn></math></p></div>


title: the-domain-of-the-function-f-x-9-x-2-s-i-n-1-x-3
slug: the-domain-of-the-function-f-x-9-x-2-s-i-n-1-x-3-14537
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">[2,3)</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The domain of the function
<br/> <br/>$f(x)=\frac{\sin ^{-1}(x-3)}{\sqrt{9-x^{2}}}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">[1,2]</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">[2,3)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">[1,2]</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">[2,3]</span> </li> </ul>

answer:
<div class="solution">$f(x)=\frac{\sin ^{-1}(x-3)}{\sqrt{9-x^{2}}}$ is defined
<br/> <br/>if (i) $-1 \leq x-3 \leq 1 \Rightarrow 2 \leq x \leq 4$ and
<br/> <br/>(ii) $9-x^{2}&gt;0 \Rightarrow-3<x></x> <br/>Taking common solution of (i) and (ii), we get $2 \leq x&lt;3$
<br/> <br/>$\therefore$ Domain $=[2,3)$</div>


title: the-domain-of-the-function-f-x-lo-g-2-lo-g-12-1-x-14-1-1-is
slug: the-domain-of-the-function-f-x-lo-g-2-lo-g-12-1-x-14-1-1-is-54136
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">(0,1)</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The domain of the function $f(x)=\log _{2}\left(-\log _{1 / 2}\left(1+\frac{1}{x^{1 / 4}}\right)-1\right)$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">(0,1)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">(0,1]</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$[1, \infty)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(1, \infty)$</span> </li> </ul>

answer:
<div class="solution">$f(x)$ is defined if
<br/> <br/>$\begin{array}{l} -\log _{1 / 2}\left(1+\frac{1}{x^{1 / 4}}\right)-1 &gt; 0 \\ \Rightarrow \quad \log _{1 / 2}\left(1+\frac{1}{x^{1 / 4}}\right) &lt; -1 \\ \Rightarrow \quad 1+\frac{1}{x^{1 / 4}} &gt; \left(\frac{1}{2}\right)^{-1} \\ \Rightarrow \quad \frac{1}{x^{1 / 4}} &gt; 1 \\ \Rightarrow \quad 0 &lt; x &lt; 1 \end{array}$</div>


title: the-domain-of-the-function-f-x-x-1-x-2-is-1
slug: the-domain-of-the-function-f-x-x-1-x-2-is-1-13068
question: <div class="question">The domain of the function $f(x)=\sqrt{x-\sqrt{1-x^2}}$ is</div>
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[-1,-\frac{1}{\sqrt{2}}\right] \cup\left[\frac{1}{\sqrt{2}}, 1\right]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$[-1,1]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(-\infty,-\frac{1}{2}\right] \cup\left[\frac{1}{\sqrt{2}},+\infty\right)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\left[\frac{1}{\sqrt{2}}, 1\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\frac{1}{\sqrt{2}}, 1\right]$</span> </div>

answer:
<div class="solution">For $f(x)$ to be defined, we must have<br/>$x-\sqrt{1-x^2} \geq 0 \text { or } x \geq \sqrt{1-x^2}&gt;0 \text { or } x^2 \geq 1-x^2$<br/>or $x^2 \geq \frac{1}{2}$.<br/>Now, $x^2 \geq \frac{1}{2} \Rightarrow\left(x-\frac{1}{\sqrt{2}}\right)\left(x+\frac{1}{\sqrt{2}}\right) \geq 0$<br/>$\Rightarrow x \leq-\frac{1}{\sqrt{2}} \quad \text { or } \quad x \geq \frac{1}{\sqrt{2}}$<br/>Also, $1-x^2 \geq 0$ or $x^2 \leq 1$.<br/>Also, $x^2 \leq 1 \Rightarrow(x-1)(x+1) \leq 0 \Rightarrow-1 \leq x \leq 1$<br/>Thus, $x&gt;0, x^2 \geq \frac{1}{2}$ and $x^2 \leq 1 \Rightarrow x \in\left[\frac{1}{\sqrt{2}}, 1\right]$</div>


title: the-domain-of-the-function-f-x-x-1-x-2-is
slug: the-domain-of-the-function-f-x-x-1-x-2-is-39406
question: <div class="question">The domain of the function $f(x)=\sqrt{x-\sqrt{1-x^{2}}}$ is</div>
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[-1,-\frac{1}{\sqrt{2}}\right] \cup\left[\frac{1}{\sqrt{2}}, 1\right]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">[-1,1]</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(-\infty,-\frac{1}{2}\right] \cup\left[\frac{1}{\sqrt{2}},+\infty\right)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\left[\frac{1}{\sqrt{2}}, 1\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\frac{1}{\sqrt{2}}, 1\right]$</span> </div>

answer:
<div class="solution">For $f(x)$ to be defined, we must have $x-\sqrt{1-x^{2}} \geq 0$ or $\quad x \geq \sqrt{1-x^{2}}&gt;0$
<br/> <br/>$\therefore \quad x^{2} \geq 1-x^{2}$ or $x^{2} \geq \frac{1}{2}$
<br/> <br/>Also, $1-x^{2} \geq 0$ or $x^{2} \leq 1$.
<br/> <br/>Now, $x^{2} \geq \frac{1}{2} \Rightarrow\left(x-\frac{1}{\sqrt{2}}\right)\left(x+\frac{1}{\sqrt{2}}\right) \geq 0$
<br/> <br/>$\Rightarrow x \leq-\frac{1}{\sqrt{2}}$ or $x \geq \frac{1}{\sqrt{2}}$
<br/> <br/>Also, $x^{2} \leq 1 \Rightarrow(x-1)(x+1) \leq 0$
<br/> <br/>$\Rightarrow-1 \leq x \leq 1$
<br/> <br/>Thus, $x&gt;0, x^{2} \geq \frac{1}{2}$ and $x^{2} \leq 1$
<br/> <br/>$\Rightarrow x \in\left[\frac{1}{\sqrt{2}}, 1\right]$</div>


title: the-domain-of-the-function-f-x-x-12-x-9-x-4-x-1-1-is-given-by
slug: the-domain-of-the-function-f-x-x-12-x-9-x-4-x-1-1-is-given-by-57637
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
question:
<div class="question">The domain of the function $\mathrm{f}(x)=\frac{1}{\sqrt{x^{12}-x^{9}+x^{4}-x+1}}$ is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(-\infty,-1)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(1, \infty)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(-1,1)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$(-\infty, \infty)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(-\infty, \infty)$</span> </div>

answer:
<div class="solution">$\mathrm{f}(x)$ is defined for $x^{12}-x^{9}+x^{4}-x+1&gt;0$
<br/> <br/>$\begin{array}{l}
<br/> <br/>\Rightarrow x^{4}\left(x^{8}+1\right)-x\left(x^{8}+1\right)+1&gt;0 \\
<br/> <br/>\Rightarrow\left(x^{8}+1\right) x\left(x^{3}-1\right)+1&gt;0
<br/> <br/>\end{array}$
<br/> <br/>If $x \geq 1$ or $x \leq-1$, then the above expres - sion is positive.
<br/> <br/>If $-1 &lt; x \leq 0$, the above inequality still holds.
<br/> <br/>If $0 &lt; x &lt; 1$, then
<br/> <br/>$\begin{array}{l}
<br/> <br/>x^{12}-x\left(x^{8}+\right)+\left(x^{4}+1\right)&gt;0 \\
<br/> <br/>\because\left[x^{4}+1&gt;x^{8}+1 \text { and so } x^{4}+1&gt;x\left(x^{8}+1\right)\right]
<br/> <br/>\end{array}$
<br/> <br/>The domain of $f=(-\infty, \infty)$</div>


tags: ['Physics', 'Current Electricity', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
title:
the-drift-velocity-of-electrons-in-silver-wire-with-crosssectional-area-314-1-0-6-m-2-carrying-a-current-of-20-a-is-given-atomic-weight-of-ag-108-dens

slug:
the-drift-velocity-of-electrons-in-silver-wire-with-crosssectional-area-314-1-0-6-m-2-carrying-a-current-of-20-a-is-given-atomic-weight-of-ag-108-dens-29207

question:
<div class="question">The drift velocity of electrons in silver wire with cross-sectional area $3.14 \times 10^{-6} \mathrm{~m}^{2}$ carrying a current of $20 \mathrm{~A}$ is. Given atomic weight of $\mathrm{Ag}=$ $108,$ density of silver $=10.5 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.798 \times 10^{-4} \mathrm{~m} / \mathrm{sec}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$67.98 \times 10^{-4} \mathrm{~m} / \mathrm{sec}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.67 \times 10^{-4} \mathrm{~m} / \mathrm{sec}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$6.798 \times 10^{-4} \mathrm{~m} / \mathrm{sec}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6.798 \times 10^{-4} \mathrm{~m} / \mathrm{sec}$</span> </div>

answer:
<div class="solution">$\quad$ Number of electrons per kg of silver
<br/> <br/>$$
<br/> <br/>=\frac{6.023 \times 10^{26}}{108}
<br/> <br/>$$
<br/> <br/>Number of electrons per unit volume of silver
<br/> <br/>$$
<br/> <br/>\begin{array}{c}
<br/> <br/>n=\frac{6.023 \times 10^{26}}{108} \times 10.5 \times 10^{3} \\
<br/> <br/>\qquad \begin{array}{c}
<br/> <br/>v_{d}=\frac{I}{n e A} \\
<br/> <br/>=\frac{20}{6.023 \times 10^{26} \times 10.5 \times 10^{3} \times 1.6 \times 10^{-19} \times 3.14 \times 10^{-6}} \times 108 \\
<br/> <br/>=6.798 \times 10^{-4} \mathrm{~m} / \mathrm{sec} .
<br/> <br/>\end{array}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


title: the-drug-that-is-antipyretic-as-well-as-analgesic-is
slug: the-drug-that-is-antipyretic-as-well-as-analgesic-is-13335
question: <div class="question">The drug that is antipyretic as well as analgesic is</div>
tags: ['Chemistry', 'Chemistry in Everyday Life', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">chloroquine</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">penicillin</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">paracetamol</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">chloropromazine hydrochloride</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">paracetamol</span> </div>

answer:
<div class="solution"><p>Acetamidophenol is an antipyretic as well as analgesic drug. It is commonly known by the name of paracetamol. It lowers body temperature as well as is used to get relief from pain. </p><p>So, the drug which acts as both antipyretic and analgesic is Acetamidophenol.</p><p>Its structure is as follows</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/0865379c-8bec-4472-852e-a63743debf65-image.png" style="width: 177px; height: 270px;"/></p></div>


title: the-drug-used-as-an-antidepressant-is
slug: the-drug-used-as-an-antidepressant-is-24888
question: <div class="question">The drug used as an antidepressant is</div>
tags: ['Chemistry', 'Chemistry in Everyday Life', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
answer: <div class="solution">Tofranil is used for the treatment of antidepressant.</div>
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Luminol</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Tofranil</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Mescaline</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Sulphadiazine</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Tofranil</span> </div>


tags: ['Mathematics', 'Ellipse', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P2.db
title:
the-eccentricity-of-an-ellipse-with-its-centre-at-the-origin-is-12-if-one-of-the-directrices-is-x-4-then-the-equation-of-the-ellipse-is

slug:
the-eccentricity-of-an-ellipse-with-its-centre-at-the-origin-is-12-if-one-of-the-directrices-is-x-4-then-the-equation-of-the-ellipse-is-73083

question:
<div class="question">The eccentricity of an ellipse, with its centre at the origin, is $1 / 2$. If one of the directrices is $\mathrm{x}=4$ then the equation of the ellipse is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4 x^{2}+3 y^{2}=1$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$3 x^{2}+4 y^{2}=12$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 x^{2}+3 y^{2}=12$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 x^{2}+4 y^{2}=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 x^{2}+4 y^{2}=12$</span> </div>

answer:
<div class="solution">$\mathrm{e}=\frac{1}{2} .$ Directrix $, \mathrm{x}=\frac{\mathrm{a}}{\mathrm{e}}=4$
<br/> <br/>$$
<br/> <br/>\therefore \mathrm{a}=4 \times \frac{1}{2}=2 \quad \therefore \mathrm{b}=2 \sqrt{1-\frac{1}{4}}=\sqrt{3}
<br/> <br/>$$
<br/> <br/>Equation of ellispe is
<br/> <br/>$$
<br/> <br/>\frac{x^{2}}{4}+\frac{y^{2}}{3}=1 \Rightarrow 3 x^{2}+4 y^{2}=12
<br/> <br/>$$</div>


title: the-eccentricity-of-the-ellipse-whose-major-axis-is-three-times-the-minor-axis-is
slug: the-eccentricity-of-the-ellipse-whose-major-axis-is-three-times-the-minor-axis-is-44021
tags: ['Mathematics', 'Ellipse', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
question:
<div class="question">The eccentricity of the ellipse whose major axis is three times the minor axis is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\sqrt{2}}{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{3}}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{2 \sqrt{2}}{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2}{\sqrt{3}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2 \sqrt{2}}{3}$</span> </div>

answer:
<div class="solution">Let a be the major axis and $b$, the minor axis of the ellipse, then 3 minor axis = major axis.
<br/> <br/>$\Rightarrow 3 b=a$
<br/> <br/>Eccentricity is given by:
<br/> <br/>$\begin{array}{l}
<br/> <br/>b^{2}=a^{2}\left(1-e^{2}\right) \\
<br/> <br/>\Rightarrow b^{2}=9 b^{2}\left(1-e^{2}\right) \\
<br/> <br/>\Rightarrow \frac{1}{9}=\left(1-e^{2}\right) \\
<br/> <br/>\Rightarrow e^{2}=\frac{8}{9} \\
<br/> <br/>\Rightarrow e=\frac{2 \sqrt{2}}{3}
<br/> <br/>\end{array}$</div>


tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P2.db
title:
the-effect-of-increase-in-temperature-on-the-number-of-electrons-in-conduction-band-n-e-and-resistance-of-a-semiconductor-will-be-as-1

slug:
the-effect-of-increase-in-temperature-on-the-number-of-electrons-in-conduction-band-n-e-and-resistance-of-a-semiconductor-will-be-as-1-51296

question:
<div class="question">The effect of increase in temperature on the number of electrons in conduction band $\left(\mathrm{n}_{\mathrm{e}}\right)$ and resistance of a semiconductor will be as:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Both $\mathrm{n}_{\mathrm{e}}$ and resistance decrease</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Both $\mathrm{n}_{\mathrm{e}}$ and resistance increase</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{n}_{\mathrm{e}}$ increases, resistance decreases</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$n_e$ decreases, resistance increases</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{n}_{\mathrm{e}}$ increases, resistance decreases</span> </div>

answer:
<div class="solution">When temperature increases, more electrons excite to conduction band and hence conductivity increases, therefore resistance decreases.</div>


title: the-efficiency-of-fuel-cell-is-given-by-the-expression-is
slug: the-efficiency-of-fuel-cell-is-given-by-the-expression-is-74467
question: <div class="question">The efficiency of fuel cell is given by the expression, $\eta$ is</div>
tags: ['Chemistry', 'Electrochemistry', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\eta=-\frac{\mathrm{nFE}_{\text {cell }}}{\Delta \mathrm{H}} \times 100$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\eta=-\frac{\mathrm{nFE}_{\text {cell }}}{\Delta \mathrm{S}} \times 100$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\eta=-\frac{\mathrm{nFE}_{\text {cell }}}{\Delta \mathrm{A}} \times 100$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\eta=-\frac{\mathrm{nFE}_{\text {cell }}}{\Delta \mathrm{H}} \times 100$</span> </div>

answer:
<div class="solution">Efficiency of fuel cell is:
<br/> <br/>$\eta=\frac{-n F E_{\text {cell }}}{\Delta H} \times 100$</div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
title:
the-electric-field-intensity-just-sufficient-to-balance-the-earths-gravitational-attraction-on-an-electron-will-be-given-mass-and-charge-of-an-electro

slug:
the-electric-field-intensity-just-sufficient-to-balance-the-earths-gravitational-attraction-on-an-electron-will-be-given-mass-and-charge-of-an-electro-87902

question:
<div class="question">The electric field intensity just sufficient to balance the earth's gravitational attraction on an electron will be: (given mass and charge of an electron respectively are $9.1 \times 10^{-31} \mathrm{~kg}$ and $1.6 \times$ $\left.10^{-19} \mathrm{C} .\right)$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$-5.6 \times 10^{-11} \mathrm{~N} / \mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-4.8 \times 10^{-15} \mathrm{~N} / \mathrm{C}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-1.6 \times 10^{-19} \mathrm{~N} / \mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-3.2 \times 10^{-19} \mathrm{~N} / \mathrm{C}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-5.6 \times 10^{-11} \mathrm{~N} / \mathrm{C}$</span> </div>

answer:
<div class="solution">$-\mathrm{eE}=\mathrm{mg}$
<br/>$\overrightarrow{\mathrm{E}}=-\frac{9.1 \times 10^{-31} \times 10}{1.6 \times 10^{-19}}=-5.6 \times 10^{-11} \mathrm{~N} / \mathrm{C}$</div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
title:
the-electric-potential-v-is-given-as-a-function-of-distance-metre-by-v-5-x-2-10-x-4-volt-question-value-of-electric-field-at-x-1-m-is

slug:
the-electric-potential-v-is-given-as-a-function-of-distance-metre-by-v-5-x-2-10-x-4-volt-question-value-of-electric-field-at-x-1-m-is-60338

question:
<div class="question">The electric potential $V$ is given as a function of distance $\times($ metre $)$ by $V=\left(5 x^{2}+10 x-4\right)$ volt.
<br/> <br/><strong>Question:</strong> Value of electric field at $x=1 \mathrm{~m}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-23 \mathrm{~V} / \mathrm{m}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$11 \mathrm{~V} / \mathrm{m}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$6 \mathrm{~V} / \mathrm{m}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$-20 \mathrm{~V} / \mathrm{m}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-20 \mathrm{~V} / \mathrm{m}$</span> </div>

answer:
<div class="solution">$\mathrm{V}=5 x^{2}+10 x-4$
<br/> <br/>$\mathrm{E}=\frac{-\mathrm{dV}}{\mathrm{dx}}=-(10 x+10)$
<br/> <br/>At $x=1 m, \mathrm{E}=-20 \mathrm{~V} / m$</div>


tags: ['Chemistry', 'Electrochemistry', 'BITSAT', 'BITSAT 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3: 2</span> </div>
source_db: MarksBatch2_P2.db
title:
the-electrolytic-cells-one-containing-acidified-ferrous-chloride-and-another-acidified-ferric-chloride-are-connected-in-series-the-ratio-of-iron-depos

slug:
the-electrolytic-cells-one-containing-acidified-ferrous-chloride-and-another-acidified-ferric-chloride-are-connected-in-series-the-ratio-of-iron-depos-59882

question:
<div class="question">The electrolytic cells, one containing acidified ferrous chloride and another acidified ferric chloride are connected in series. The ratio of iron deposited at cathodes in the two cells when electricity is passed through the cells will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">3: 1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2: 1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1: 1</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">3: 2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">At cathodes: $\mathrm{Fe}^{+}+2 \mathrm{e}^{-} \longrightarrow \mathrm{Fe} ;$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\mathrm{Fe}^{3+}+3 \mathrm{e}^{-} \longrightarrow \mathrm{Fe} \\
<br/> <br/>\left(\mathrm{E}_{\mathrm{Fe}}\right)_{1}=\frac{\text { At. wt. }}{2} ;\left(\mathrm{E}_{\mathrm{Fe}}\right)_{2}=\frac{\text { At. wt. }}{3}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Hence, $\frac{\left(\mathrm{E}_{\mathrm{Fe}}\right)_{1}}{\left(\mathrm{E}_{\mathrm{Fe}}\right)_{2}}=\frac{3}{2}$</div>


title: the-element-with-atomic-number-118-will-be
slug: the-element-with-atomic-number-118-will-be-11135
question: <div class="question">The element, with atomic number $118,$ will be</div>
tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">alkali</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">noble gas</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">lanthanide</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">transition element</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">noble gas</span> </div>

answer:
<div class="solution">Electronic configuration of element with atomic number 118 will be $[\mathrm{Rn}] 5 \mathrm{f}^{14} 6 \mathrm{~d}^{10}$ $7 \mathrm{~s}^{2} 7 \mathrm{p}^{6} .$ Since its elctronic configuration in the outer moot or bit (ns $^{2} n p^{6}$ ) resemble with that of inert or noble gases, therefore it will be noble gas element.</div>


title: the-emf-of-the-cell-tl-tl-0001-m-cu-2-001-m-cu-is-083-the-cell-emf-can-be-increased-by
slug: the-emf-of-the-cell-tl-tl-0001-m-cu-2-001-m-cu-is-083-the-cell-emf-can-be-increased-by-81229
tags: ['Chemistry', 'Electrochemistry', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
question:
<div class="question">The EMF of the cell $\mathrm{Tl} / \mathrm{Tl}^{+}(0.001 \mathrm{M}) \| \mathrm{Cu}^{2+}$
<br/> <br/>$(0.01 \mathrm{M}) / \mathrm{Cu}$ is $0.83 .$ The cell EMF can be increased by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Increasing the concentration of $\mathrm{Tl}^{+}$ ions.</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Increasing the concentration of $\mathrm{Cu}^{2+}$ ions.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Increasing the concentration of $\mathrm{Tl}^{+}$ and $\mathrm{Cu}^{2+}$ ions.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Increasing the concentration of $\mathrm{Cu}^{2+}$ ions.</span> </div>

answer:
<div class="solution">The oxidation potential
<br/> <br/>$\propto \frac{1}{\text { Concentration of ions }}$ and reduction
<br/> <br/>potential $\propto$ concentration of ions. The cell voltage can be increased by decreasing the concentration of ions around anode or by increasing the concentration of ions around cathode</div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
title:
the-empirical-formula-and-molecular-mass-of-a-compound-are-ch-2-o-and-180-g-respectively-the-molecular-formula-of-the-compound-will-be

slug:
the-empirical-formula-and-molecular-mass-of-a-compound-are-ch-2-o-and-180-g-respectively-the-molecular-formula-of-the-compound-will-be-39545

question:
<div class="question">The empirical formula and molecular mass of a compound are <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>180</mn><mo> </mo><mi mathvariant="normal">g</mi></math> respectively. The molecular formula of the compound will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mn>9</mn></msub><msub><mi mathvariant="normal">H</mi><mn>18</mn></msub><msub><mi mathvariant="normal">O</mi><mn>9</mn></msub></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mn>6</mn></msub><msub><mi mathvariant="normal">H</mi><mn>12</mn></msub><msub><mi mathvariant="normal">O</mi><mn>6</mn></msub></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mn>2</mn></msub><msub><mi mathvariant="normal">H</mi><mn>4</mn></msub><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mn>6</mn></msub><msub><mi mathvariant="normal">H</mi><mn>12</mn></msub><msub><mi mathvariant="normal">O</mi><mn>6</mn></msub></math></span> </div>

answer:
<div class="solution"><p>As we know that, the Molar mass of Carbon=12</p><p>Molar mass of Hydrogen=1</p><p>Molar mass of Oxygen=16</p><p>Empirical formula mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msub><mi>CH</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo>=</mo><mn>12</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>×</mo><mn>16</mn><mo>=</mo><mn>30</mn></math></p><p>Molecular mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>180</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mfrac><mtext> Molccular Mass </mtext><mtext> Empirical formula mass </mtext></mfrac><mo>=</mo><mfrac><mn>100</mn><mn>30</mn></mfrac><mo>=</mo><mn>6</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo></math> Molecular formula <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">n</mi><mo>×</mo></math> empirical formula</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Molecular formula <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>6</mn><mo>×</mo><msub><mi>CH</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo>=</mo><msub><mi mathvariant="normal">C</mi><mn>6</mn></msub><msub><mi mathvariant="normal">H</mi><mn>12</mn></msub><msub><mi mathvariant="normal">O</mi><mn>6</mn></msub></math></p></div>


tags: ['Physics', 'Atomic Physics', 'BITSAT', 'BITSAT 2011']
answer: <div class="solution">No Solution available</div>
source_db: MarksBatch2_P2.db
title:
the-energies-of-energy-levels-a-b-and-c-for-a-given-atom-are-in-the-sequence-e-a-e-b-e-c-if-the-radiations-of-wavelengths-1-2-and-3-are-emitted-due-to

slug:
the-energies-of-energy-levels-a-b-and-c-for-a-given-atom-are-in-the-sequence-e-a-e-b-e-c-if-the-radiations-of-wavelengths-1-2-and-3-are-emitted-due-to-93193

question:
<div class="question">The energies of energy levels $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ for a given atom are in the sequence $\mathrm{E}_{\mathrm{A}}&lt;\mathrm{E}_{\mathrm{B}}&lt;\mathrm{E}_{\mathrm{C}}$. If<br/>the radiations of wavelengths $\lambda_{1}, \lambda_{2}$ and $\lambda_{3}$ are emitted due to the atomic transitions $\mathrm{C}$ to $\mathrm{B}, \mathrm{B}$ to A and $C$ to A respectively then which of the following relations is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\lambda_{1}+\lambda_{2}+\lambda_{3}=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\lambda_{3}=\lambda_{1}^{2}+\lambda_{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\lambda_{3}=\lambda_{1}+\lambda_{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\lambda_{3}=\frac{\lambda_{1} \lambda_{2}}{\lambda_{1}+\lambda_{2}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\lambda_{3}=\frac{\lambda_{1} \lambda_{2}}{\lambda_{1}+\lambda_{2}}$</span> </div>


tags: ['Physics', 'Dual Nature of Matter', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">D</span> </div>
source_db: MarksBatch2_P2.db
title:
the-energy-levels-of-an-atom-is-shown-is-figure-which-one-of-these-transitions-will-result-in-the-emission-of-a-photon-of-wavelength-1241-nm-given-h-6

slug:
the-energy-levels-of-an-atom-is-shown-is-figure-which-one-of-these-transitions-will-result-in-the-emission-of-a-photon-of-wavelength-1241-nm-given-h-6-26995

question:
<div class="question">The energy levels of an atom is shown is figure. Which one of these transitions will result in the emission of a photon of wavelength $124.1 \mathrm{~nm}$ ? Given $\left(\mathrm{h}=6.62 \times 10^{-34} \mathrm{Js}\right)$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/hcgomixBL6ft6T8i46EY22HXi9sQB6M3nDhUsife694.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">B</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">A</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">C</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">D</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">As $\mathrm{E}(\mathrm{eV})=\frac{1240}{\lambda(\mathrm{nm})}=\frac{1240}{124.1} \simeq 10 \mathrm{eV}$<br/>Only is transition (D), the energy gap is $10 \mathrm{eV}$ So, option (d) is correct</div>


tags: ['Physics', 'Atomic Physics', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
title:
the-energy-of-electron-in-the-nth-orbit-of-hydrogen-atom-is-expressed-as-e-n-n-2-136-ev-the-shortest-and-longest-wavelength-of-lyman-series-will-be

slug:
the-energy-of-electron-in-the-nth-orbit-of-hydrogen-atom-is-expressed-as-e-n-n-2-136-ev-the-shortest-and-longest-wavelength-of-lyman-series-will-be-90895

question:
<div class="question">The energy of electron in the nth orbit of hydrogen atom is expressed as $\mathrm{E}_{\mathrm{n}}=\frac{-13.6}{\mathrm{n}^{2}} \mathrm{eV}$.
<br/> <br/>The shortest and longest wavelength of Lyman series will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$910 Ã, 1213 Ã$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5463 Ã, 7858 Ã$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1315 Ã, 1530 Ã$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$910 Ã, 1213 Ã$</span> </div>

answer:
<div class="solution">$\frac{1}{\lambda_{\max }}=\mathrm{R}\left[\frac{1}{(\mathrm{l})^{2}}-\frac{1}{(2)^{2}}\right]$
<br/> <br/>$\Rightarrow \lambda_{\mathrm{max}}=\frac{4}{3 \mathrm{R}} \approx 1213 Ãâ¦$
<br/> <br/>and
<br/> <br/>$\frac{1}{\lambda_{\min }}=\mathrm{R}\left[\frac{1}{(1)^{2}}-\frac{1}{\infty}\right] \Rightarrow \lambda_{\min }=\frac{1}{\mathrm{R}} \approx 910 Ãâ¦$</div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P2.db
title:
the-engine-of-a-train-moving-with-speed-10-ms-1-towards-a-platform-sounds-a-whistle-at-frequency-400-hz-the-frequency-heard-by-a-passenger-inside-the-

slug:
the-engine-of-a-train-moving-with-speed-10-ms-1-towards-a-platform-sounds-a-whistle-at-frequency-400-hz-the-frequency-heard-by-a-passenger-inside-the-69201

question:
<div class="question">The engine of a train moving with speed $10 \mathrm{~ms}^{-1}$ towards a platform sounds a whistle at frequency $400 \mathrm{~Hz}$. The frequency heard by a passenger inside the train is (neglect air speed. Speed of sound in air $330 \mathrm{~ms}^{-1}$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$200 \mathrm{~Hz}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$400 \mathrm{~Hz}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$412 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$388 \mathrm{~Hz}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$400 \mathrm{~Hz}$</span> </div>

answer:
<div class="solution">The relative velocity of a passenger with respect to source of sound (engine of train) is 0 . So there will be no Doppler's effect. So frequency heard is $400 \mathrm{~Hz}$.</div>


tags: ['Physics', 'Units and Dimensions', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
title:
the-engine-of-a-truck-moving-along-a-straight-road-delivers-constant-power-the-distance-travelled-by-the-truck-in-time-t-is-proportional-to

slug:
the-engine-of-a-truck-moving-along-a-straight-road-delivers-constant-power-the-distance-travelled-by-the-truck-in-time-t-is-proportional-to-71207

question:
<div class="question">The engine of a truck moving along a straight road delivers constant power. The distance travelled by the truck in time $t$ is proportional to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$t$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$t^{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{\mathrm{t}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$t^{3 / 2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$t^{3 / 2}$</span> </div>

answer:
<div class="solution">$\frac{\mathrm{mv}^{2}}{\mathrm{t}}=\frac{\mathrm{ms}^{2}}{\mathrm{t}^{3}}$
<br/> <br/>since both $\mathrm{P}$ and $\mathrm{m}$ are constants
<br/> <br/>$\therefore \frac{\mathrm{s}^{2}}{\mathrm{t}^{3}}=$ constant</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P2.db
title:
the-enthalpy-of-combustion-of-2-moles-of-benzene-at-2-7-c-differs-from-the-value-determined-in-bomb-calorimeter-by

slug:
the-enthalpy-of-combustion-of-2-moles-of-benzene-at-2-7-c-differs-from-the-value-determined-in-bomb-calorimeter-by-81382

question:
<div class="question">The enthalpy of combustion of 2 moles of benzene at $27^{\circ} \mathrm{C}$ differs from the value determined in bomb calorimeter by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-2.494 \mathrm{~kJ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.494 \mathrm{~kJ}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-7.483 \mathrm{~kJ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$7.483 \mathrm{~kJ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-7.483 \mathrm{~kJ}$</span> </div>

answer:
<div class="solution">By bomb calorimeter we get $\Delta \mathrm{E}$.
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>2 \mathrm{C}_{6} \mathrm{H}_{6}(\mathrm{l})+15 \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow \\
<br/> <br/>12 \mathrm{CO}_{2}(\mathrm{~g})+6 \mathrm{H}_{2} \mathrm{O}(1) \\
<br/> <br/>\Delta \mathrm{H}-\Delta \mathrm{E}=\Delta \mathrm{nRT} \\
<br/> <br/>=(12-15) \times 8.314 \times 300=-7.483 \mathrm{~kJ}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'BITSAT', 'BITSAT 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2735</span> </div>
source_db: MarksBatch2_P2.db
title:
the-enthalpy-of-sublimation-of-aluminium-is-330-kj-mol-its-ist-ii-nd-and-iii-rd-ionization-enthalpies-are-5801820-and-2740-kj-respectively-how-much-he

slug:
the-enthalpy-of-sublimation-of-aluminium-is-330-kj-mol-its-ist-ii-nd-and-iii-rd-ionization-enthalpies-are-5801820-and-2740-kj-respectively-how-much-he-29767

question:
<div class="question">The enthalpy of sublimation of aluminium is 330 $\mathrm{kJ} / \mathrm{mol}$. Its Ist, II $^{\text {nd }}$ and III $^{\text {rd }}$ ionization enthalpies are 580,1820 and $2740 \mathrm{~kJ}$ respectively. How much heat has too be supplied (in $\mathrm{kJ}$ ) to convert $13.5 \mathrm{~g}$ of aluminium into $\mathrm{Al}^{3+}$ ions and electrons at $298 \mathrm{k}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5470</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2735</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4105</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3765</span> </li> </ul>

answer:
<div class="solution">Heat needed too be supplied
<br/> <br/>$$
<br/> <br/>\text { per } m o l=330+580+1820+2740
<br/> <br/>$$
<br/> <br/>$=5470 \mathrm{~kJ}$
<br/> <br/>No. of mols of Al taken $=\frac{13.5}{27}=0.5 \mathrm{~mol}$
<br/> <br/>Heat required $=0.5 \times 5470 \mathrm{~kJ}=2735 \mathrm{~kJ}$</div>


title: the-equation-of-a-common-tangent-to-the-parabolas-y-x-2-and-y-x-2-2-is
slug: the-equation-of-a-common-tangent-to-the-parabolas-y-x-2-and-y-x-2-2-is-36186
tags: ['Mathematics', 'Parabola', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">The equation of a common tangent to the parabolas $y=x^2$ and $y=-(x-2)^2$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y=4(x-2)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$y=4(x-1)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$y=4(x+1)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$y=4(x+2)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y=4(x-1)$</span> </div>

answer:
<div class="solution">Equation of tangent of parabola $y=x^2$ be <br/>$t x=y+a t^2$ ...(i)<br/>$y=t x-\frac{t^2}{4}$<br/>Solve with $\mathrm{y}=-(\mathrm{x}-2)^2$<br/>$t x-\frac{t^2}{4}=-(x-2)^2$<br/>$x^2+x(t-4)-\frac{t^2}{4}+4=0$<br/>Here, Discriminant $=0$.<br/>$(t-4)^2-4 \cdot\left(4-\frac{t^2}{4}\right)=0 \Rightarrow t^2-4 t=0 \Rightarrow t=0$<br/>or $\mathrm{t}=4$<br/>Put value of $t$ in eq. (i), then $y=4(x-1)$.</div>


title: the-equation-of-a-projectile-is-y-3-x-2-g-x-2-the-angle-of-projection-is-given-by
slug: the-equation-of-a-projectile-is-y-3-x-2-g-x-2-the-angle-of-projection-is-given-by-92454
tags: ['Physics', 'Motion In Two Dimensions', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
question:
<div class="question">The equation of a projectile is $y=\sqrt{3} x-\frac{g x^{2}}{2}$ The angle of projection is given by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\tan \theta=\frac{1}{\sqrt{3}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\tan \theta=\sqrt{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">zero.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan \theta=\sqrt{3}$</span> </div>

answer:
<div class="solution">Comparing the given equation with
<br/> <br/>$\mathrm{y}=\mathrm{x} \tan \theta-\frac{\mathrm{gx}^{2}}{2 \mathrm{u}^{2} \cos ^{2} \theta}, \text { we get }$
<br/> <br/>$\tan \theta=\sqrt{3}$</div>


title: the-equation-of-all-lines-having-slope-2-which-are-tangent-to-the-curve-y-x-3-1-x-3-is
slug: the-equation-of-all-lines-having-slope-2-which-are-tangent-to-the-curve-y-x-3-1-x-3-is-44000
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P2.db
question:
<div class="question">The equation of all lines having slope 2 which are tangent to the curve $y=\frac{1}{x-3}, x \neq 3,$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y=2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y=2 x$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$y=2 x+3$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">The equation of the given curve is $\mathrm{y}=\frac{1}{\mathrm{x}-3}, \mathrm{x} \neq 3$
<br/> <br/>The slope of the tangent to the given curve
<br/> <br/>at any point $(\mathrm{x}, \mathrm{y})$ is given by $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-1}{(\mathrm{x}-3)^{2}}$
<br/> <br/>For tangent having slope $2,$ we must have
<br/> <br/>$2=\frac{-1}{(x-3)^{2}}$
<br/> <br/>$$
<br/> <br/>\Rightarrow 2(x-3)^{2}=-1 \Rightarrow(x-3)^{2}=-\frac{1}{2}
<br/> <br/>$$
<br/> <br/>which is not possible as square of a real number cannot be negative. Hence, there is no tangent to the given curve having slope 2 .</div>


title: the-equation-of-plane-containing-line-x-y-1-z-1-and-parallel-to-x-2-z-3-1-y-3-is
slug: the-equation-of-plane-containing-line-x-y-1-z-1-and-parallel-to-x-2-z-3-1-y-3-is-17990
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
question:
<div class="question">The equation of plane containing line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mi>y</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>z</mi><mo>=</mo><mn>1</mn></math> and parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mi>z</mi><mn>3</mn></mfrac><mo>=</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>=</mo><mn>3</mn></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>1</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>1</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>1</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>=</mo><mo>-</mo><mn>1</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>1</mn></math></span> </div>

answer:
<div class="solution"><p>Equation of plane containing line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mi>y</mi><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>x</mi><mo>-</mo><mi>y</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>λ</mi><mo>(</mo><mi>z</mi><mo>-</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mi>λ</mi><mi>z</mi><mo>-</mo><mn>1</mn><mo>-</mo><mi>λ</mi><mo>=</mo><mn>0</mn></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math></p><p>Since, this plane is parallel to line <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mi>z</mi><mn>3</mn></mfrac><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>-</mo><mn>2</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo>-</mo><mn>3</mn></mrow><mn>0</mn></mfrac><mo>=</mo><mfrac><mi>z</mi><mn>3</mn></mfrac></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>ii</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mn>2</mn><mo>(</mo><mn>1</mn><mo>)</mo><mo>+</mo><mo>(</mo><mn>0</mn><mo>)</mo><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>3</mn><mo>(</mo><mi>λ</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>λ</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>/</mo><mn>3</mn></math></p><p>Putting the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math> in Eq. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mi>y</mi><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mi>z</mi><mo>-</mo><mn>1</mn><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>=</mo><mn>0</mn><mspace linebreak="newline"></mspace><mo>⇒</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>1</mn></math></p></div>


title: the-equation-of-tangent-to-the-curve-ysin-x-at-the-point-pi-0-is
slug: the-equation-of-tangent-to-the-curve-ysin-x-at-the-point-pi-0-is-11857
tags: ['Mathematics', 'Indefinite Integration', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P2.db
question:
<div class="question">The equation of tangent to the curve \(y=\sin x\) at the point \((\pi, 0)\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(x+y=0\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(x+y=\pi\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(x-y=\pi\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(x-y=0\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(x+y=\pi\)</span> </div>

answer:
<div class="solution">\(y=\sin x \Rightarrow \frac{d y}{d x}=\cos x \Rightarrow\left(\frac{d y}{d x}\right)_{(\pi, 0)}=-1\)<br/>Therefore the equation of tangent at \((\pi, 0)\) is given by<br/>\(y-0=-1(x-\pi) \Rightarrow x+y=\pi\)</div>


title: the-equation-of-the-circle-which-touches-the-line-y-5-and-passes-through-1-2-and-1-2-is
slug: the-equation-of-the-circle-which-touches-the-line-y-5-and-passes-through-1-2-and-1-2-is-70337
tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
question:
<div class="question">The equation of the circle, which touches the line $y=5$ and passes through $(-1,2)$ and $(1,2)$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$9 x^{2}+9 y^{2}-60 y+75=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$9 x^{2}+9 y^{2}-60 x-75=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$9 x^{2}+9 y^{2}+60 y-75=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$9 x^{2}+9 y^{2}+60 x+75=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$9 x^{2}+9 y^{2}-60 y+75=0$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/qSv62DZUInezs9OtYo4L1dI8LmR24DBWnsy_SoSgY6A.original.fullsize.png"/><br/> <br/> <br/>The centre of the circle is on the perpendicular bisector of the line joining $(-1,2)$ and $(1,2)$, which is the $y$-axis. The ordinate of the centre is given by
<br/> <br/>$\begin{array}{l}
<br/> <br/>(5-y)^{2}=1+(y-2)^{2} \\
<br/> <br/>\Rightarrow y=\frac{10}{3}
<br/> <br/>\end{array}$
<br/> <br/>Hence, eq. of the circle is:
<br/> <br/>$\begin{array}{l}
<br/> <br/>x^{2}+\left(y-\frac{10}{3}\right)^{2}=\left(\frac{5}{3}\right)^{2} \\
<br/> <br/>\Rightarrow 9 x^{2}+9 y^{2}-60 y+75=0
<br/> <br/>\end{array}$</div>


title: the-equation-of-the-circle-with-centre-0-2-and-radius-2-is-x-2-y-2-m-y-0-the-value-of-m-is
slug: the-equation-of-the-circle-with-centre-0-2-and-radius-2-is-x-2-y-2-m-y-0-the-value-of-m-is-99019
tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value"><br/>4</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The equation of the circle with centre $(0,2)$ and radius 2 is $x^2+y^2-m y=0$. The value of $m$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><br/>4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>3</span> </li> </ul>

answer:
<div class="solution">Here $h=0, k=2$ and $r=2$. Therefore, the required equation of the circle is :<br/>$(x-0)^2+(y-2)^2=(2)^2$<br/>or $x^2+y^2-4 y+4=4$ or $x^2+y^2-4 y=0$</div>


tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
title:
the-equation-of-the-curve-whose-slope-at-any-point-is-equal-to-y-2-x-and-which-passes-through-the-origin-is

slug:
the-equation-of-the-curve-whose-slope-at-any-point-is-equal-to-y-2-x-and-which-passes-through-the-origin-is-42531

question:
<div class="question">The equation of the curve whose slope at any point is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi></math> and which passes through the origin is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><mn>1</mn></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>e</mi><mi>x</mi></msup><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p>We have, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>-</mo><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi></math></p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>IF</mi><mo>=</mo><msup><mi>e</mi><mrow><mo>-</mo><mo>∫</mo><mn>1</mn><mi>d</mi><mi>x</mi></mrow></msup><mo>=</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>·</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mo>=</mo><mo>∫</mo><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>+</mo><mi>k</mi><mo>,</mo><mo> </mo><mi>k</mi></math> be the constant of integration</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfenced close="]" open="["><mrow><mi>x</mi><mo>∫</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>-</mo><mo>∫</mo><mn>1</mn><mo>·</mo><mfenced><mrow><mo>-</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></mrow></mfenced><mi>d</mi><mi>x</mi></mrow></mfenced><mo>+</mo><mi>k</mi></math>  [using integration by parts]</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>y</mi><mo>·</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mo>-</mo><mn>2</mn><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mo>+</mo><mi>k</mi><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p>As curve (i) passes through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mn>0</mn><mo>=</mo><mn>0</mn><mo>-</mo><mn>2</mn><mo>+</mo><mi>k</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>k</mi><mo>=</mo><mn>2</mn></math></p><p>Thus, the curve is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mo>-</mo><mn>2</mn><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mo>+</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>e</mi><mi>x</mi></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced></math></p><p>Hence, option (b) is correct.</p></div>


title: the-equation-of-the-ellipse-with-focus-at-pm-50-and-xfrac365-as-one-directrix-is
slug: the-equation-of-the-ellipse-with-focus-at-pm-50-and-xfrac365-as-one-directrix-is-85188
tags: ['Mathematics', 'Ellipse', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P2.db
question:
<div class="question">The equation of the ellipse with focus at \(( \pm 5,0)\) and \(x=\frac{36}{5}\) as one directrix is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{x^2}{36}+\frac{y^2}{25}=1\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\frac{x^2}{36}+\frac{y^2}{11}=1\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{x^2}{25}+\frac{y^2}{11}=1\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{x^2}{36}+\frac{y^2}{11}=1\)</span> </div>

answer:
<div class="solution">We have ae \(=5\) [Since focus is \(( \pm \mathrm{ae}, 0)\)]<br/>and \(\frac{\mathrm{a}}{\mathrm{e}}=\frac{36}{5} \quad\left[\right.\) since directrix is \(\left.\mathrm{x}= \pm \frac{\mathrm{a}}{\mathrm{e}}\right]\)<br/>On solving we get \(\mathrm{a}=6\)<br/>\(\text { and } \mathrm{e}=\frac{5}{6} \Rightarrow \mathrm{b}^2=\mathrm{a}^2\left(1-\mathrm{e}^2\right)=36\left(1-\frac{25}{36}\right)=11\)<br/>Thus, the required equation of the ellipse is<br/>\(\frac{x^2}{36}+\frac{y^2}{11}=1\)</div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P2.db
title:
the-equation-of-the-plane-containing-the-line-fracmathrmxmathrmx1ellfracmathrmymathrmy1mathrmmfracmathrmzmathrmz1mathrmn-is-aleftxx1rightbleftyy1right

slug:
the-equation-of-the-plane-containing-the-line-fracmathrmxmathrmx1ellfracmathrmymathrmy1mathrmmfracmathrmzmathrmz1mathrmn-is-aleftxx1rightbleftyy1right-13657

question:
<div class="question">The equation of the plane containing the line \(\frac{\mathrm{x}-\mathrm{x}_1}{\ell}=\frac{\mathrm{y}-\mathrm{y}_1}{\mathrm{~m}}=\frac{\mathrm{z}-\mathrm{z}_1}{\mathrm{n}}\) is \(a\left(x-x_1\right)+b\left(y-y_1\right)+c\left(z-z_1\right)=0\), then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{ax}_1+\mathrm{by}_1+\mathrm{cz}_1=0\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(a \ell+b m+c n=0\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{\mathrm{a}}{\ell}=\frac{\mathrm{b}}{\mathrm{m}}=\frac{\mathrm{c}}{\mathrm{n}}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\ell \mathrm{x}_1+\mathrm{my}_1+\mathrm{nz}_1=0\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(a \ell+b m+c n=0\)</span> </div>

answer:
<div class="solution">If the given plane contains the given line then the normal to the plane, must be perpendicular to the line and the condition for the same is \(a \ell+b m+c n=0\).</div>


title: the-equation-of-the-right-bisector-plane-of-the-segment-joining-234-and-678-is
slug: the-equation-of-the-right-bisector-plane-of-the-segment-joining-234-and-678-is-57967
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P2.db
question:
<div class="question">The equation of the right bisector plane of the segment joining (2,3,4) and (6,7,8) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x+y+z+15=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x+y+z-15=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x-y+z-15=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x+y+z-15=0$</span> </div>

answer:
<div class="solution">If the given points be $A(2,3,4)$ and $B$ $(6,7,8),$ then their mid-point $\mathrm{N}(4,5,6)$ must lie on the plane. The direction ratios of $\mathrm{AB}$ are $4,4,4,$ i.e. 1,1,1
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/FWpePhV8gc4X4C6HfEhqdq70nJpMuPNrtv7yGm7p36k.original.fullsize.png"/><br/> <br/> <br/>$\therefore$ The required plane passes through $\mathrm{N}(4,5,6)$ and is normal to $\mathrm{AB}$. Thus its equation is
<br/> <br/>$$
<br/> <br/>1(x-4)+1(y-5)+1(z-6)=0 \Rightarrow x+y+z=15
<br/> <br/>$$</div>


title: the-equation-of-the-tangent-to-the-curve-y-b-e-x-a-at-the-point-where-it-crosses-the-y-axis-is
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
slug:
the-equation-of-the-tangent-to-the-curve-y-b-e-x-a-at-the-point-where-it-crosses-the-y-axis-is-44272

question:
<div class="question">The equation of the tangent to the curve $y=b e^{-x/a}$ at the point where it crosses the $y$ -axis is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{x}{a}-\frac{y}{b}=1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a x+b y=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a x-b y=1$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{x}{a}+\frac{y}{b}=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x}{a}+\frac{y}{b}=1$</span> </div>

answer:
<div class="solution">$y=b e^{-x / a}$ meets the $y$ -axis at $(0, b)$. Again,
<br/> <br/>$\frac{d y}{d x}=b e^{-x / a}\left(-\frac{1}{a}\right)$
<br/> <br/>$\operatorname{At}(0, b), \frac{d y}{d x}=b e^{0}\left(-\frac{1}{a}\right)=-\frac{b}{a}$
<br/> <br/>Therefore, required tangent is
<br/> <br/>$y-b=-\frac{b}{a}(x-0) \text { or } \frac{x}{a}+\frac{y}{b}=1$</div>


title: the-equation-sin-4-x-k-2-sin-2-x-k-3-0-possesses-a-solution-if
slug: the-equation-sin-4-x-k-2-sin-2-x-k-3-0-possesses-a-solution-if-63114
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
question:
<div class="question">The equation $\sin ^{4} x-(k+2) \sin ^{2} x-(k+3)=0$ possesses a solution if</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$k&gt;-3$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$k&lt;-2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-3 \leq \mathrm{k} \leq-2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{k}$ is any positive integer</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-3 \leq \mathrm{k} \leq-2$</span> </div>

answer:
<div class="solution">We have, $\sin ^{4} x-(k+2) \sin ^{2} x-(k+3)=0$
<br/> <br/>$\Rightarrow \sin ^{2} x=\frac{(k+2) \pm \sqrt{(k+2)^{2}+4(k+3)}}{2}$
<br/> <br/>$=\frac{(k+2) \pm(k+4)}{2}$
<br/> <br/>$\Rightarrow \sin ^{2} x=k+3\left(\because \sin ^{2} x=-1\right.$ is not possible)
<br/> <br/>Since $0 \leq \sin ^{2} x \leq 1$
<br/> <br/>$\therefore 0 \leq \mathrm{k}+3 \leq 1$ or $-3 \leq \mathrm{k} \leq-2$</div>


title: the-equation-x-2-2-3-x-y-3-y-2-3-x-3-3-y-4-0-represents
slug: the-equation-x-2-2-3-x-y-3-y-2-3-x-3-3-y-4-0-represents-45323
tags: ['Mathematics', 'Pair of Lines', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
question:
<div class="question">The equation $x^{2}-2 \sqrt{3} x y+3 y^{2}-3 x+3 \sqrt{3} y-4=0 \quad$ represents</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a pair of intersecting lines</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">a pair of parallel lines with distance between
<br/> <br/>them $\frac{5}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">a pair of parallel lines with distance between them $5 \sqrt{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">a conic section, which is not a pair of straight lines</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a pair of parallel lines with distance between
<br/> <br/>them $\frac{5}{2}$</span> </div>

answer:
<div class="solution">We have $a=1, h=-\sqrt{3}, b=3, g=-\frac{3}{2}$, $\mathrm{f}=\frac{3 \sqrt{3}}{2}, \mathrm{c}=-4$
<br/> <br/>Thus $a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0$
<br/> <br/>Hence the equation represents a pair of straight lines.
<br/> <br/>Again $\frac{\mathrm{a}}{\mathrm{h}}=\frac{\mathrm{h}}{\mathrm{b}}=\frac{\mathrm{g}}{\mathrm{f}}=-\frac{1}{\sqrt{3}}$
<br/> <br/>$\therefore$ the lines are parallel. The distance between them
<br/> <br/>$$
<br/> <br/>=2 \sqrt{\frac{g^{2}-a c}{a(a+b)}}=2 \sqrt{\frac{\frac{9}{4}+4}{1(1+3)}}=\frac{5}{2}
<br/> <br/>$$</div>


title: the-equations-2-x-3-y-4-0-3-x-4-y-6-0-and-4-x-5-y-8-0-are
slug: the-equations-2-x-3-y-4-0-3-x-4-y-6-0-and-4-x-5-y-8-0-are-86593
question: <div class="question">The equations $2 x+3 y+4=0 ; 3 x+4 y+6=0$ and $4 x+5 y+8=0$ are</div>
tags: ['Mathematics', 'Determinants', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">consistent with unique solution</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">inconsistent</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">consistent with infinitely many solutions</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">consistent with unique solution</span> </div>

answer:
<div class="solution">Consider first two equations:
<br/>$2 x+3 y=-4 \quad$ and $\quad 3 x+4 y=-6$
<br/>We have $\Delta=\left|\begin{array}{ll}2 &amp; 3 \\ 3 &amp; 4\end{array}\right|=-1 \neq 0$
<br/>$$
<br/>\Delta_{\mathrm{x}}=\left|\begin{array}{ll}
<br/>-4 &amp; 3 \\
<br/>-6 &amp; 4
<br/>\end{array}\right|=2 \quad \text { and } \quad \Delta_{\mathrm{y}}=\left|\begin{array}{cc}
<br/>2 &amp; -4 \\
<br/>3 &amp; -6
<br/>\end{array}\right|=0
<br/>$$
<br/>$\therefore x=-2$ and $y=0$
<br/>Now this solution satisfies the third, so the equations are consistent with unique solution.</div>


tags: ['Chemistry', 'Electrochemistry', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
title:
the-equilibrium-constant-for-the-disproportionation-reaction-2-cu-aq-cu-s-cu-2-ag-at-2-5-c-e-cu-cu-052-v-e-cu-2-cu-016-v-is

slug:
the-equilibrium-constant-for-the-disproportionation-reaction-2-cu-aq-cu-s-cu-2-ag-at-2-5-c-e-cu-cu-052-v-e-cu-2-cu-016-v-is-54645

question:
<div class="question">The equilibrium constant for the disproportionation reaction $2 \mathrm{Cu}^{+}(\mathrm{aq}) \longrightarrow \mathrm{Cu}(\mathrm{s})+\mathrm{Cu}^{2+}(\mathrm{ag})$ at $25^{\circ} \mathrm{C}$
<br/> <br/>$\left(\mathrm{E}^{\circ} \mathrm{Cu}^{+} / \mathrm{Cu}=0.52 \mathrm{~V}, \mathrm{E}^{\circ} \mathrm{Cu}^{2+} / \mathrm{Cu}=0.16 \mathrm{~V}\right)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$6 \times 10^{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$6 \times 10^{6}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$1.2 \times 10^{6}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1.2 \times 10^{-6}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.2 \times 10^{6}$</span> </div>

answer:
<div class="solution">The reaction
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>2 \mathrm{Cu}^{+}(\mathrm{aq}) \longrightarrow \mathrm{Cu}(\mathrm{s})+\mathrm{Cu}^{2+}(\mathrm{aq}) \\
<br/> <br/>\mathrm{E}_{\mathrm{cell}}=\mathrm{E}_{\mathrm{cell}}^{\circ}-\frac{0.0592}{1} \log \frac{\left[\mathrm{Cu}^{2+}\right]}{\left[\mathrm{Cu}^{+}\right]^{2}}
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>At equilibrium $\mathrm{E}_{\mathrm{cell}}=0$
<br/> <br/>$$
<br/> <br/>\therefore \quad \mathrm{E}_{\text {cell }}^{\circ}=0.0592 \log \mathrm{K}_{\mathrm{c}}
<br/> <br/>$$
<br/> <br/>or, $\quad \log \mathrm{K}_{\mathrm{c}}=\frac{0.52-0.16}{0.0592}$
<br/> <br/>$\therefore \mathrm{K}_{\mathrm{c}}=1.2 \times 10^{6}$</div>


tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
title:
the-escape-velocity-from-a-planet-is-v-e-a-tunnel-is-dug-along-a-diameter-of-the-planet-and-a-small-body-is-dropped-into-it-at-the-surface-when-the-bo

slug:
the-escape-velocity-from-a-planet-is-v-e-a-tunnel-is-dug-along-a-diameter-of-the-planet-and-a-small-body-is-dropped-into-it-at-the-surface-when-the-bo-97589

question:
<div class="question">The escape velocity from a planet is $v_{e}$. A tunnel is dug along a diameter of the planet and a small body is dropped into it at the surface. When the body reaches the centre of the planet, its speed will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{V}_{\mathrm{e}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{v}_{\mathrm{e}} / \sqrt{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{v}_{\mathrm{e}} / 2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">zero</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{v}_{\mathrm{e}} / \sqrt{2}$</span> </div>

answer:
<div class="solution">$\frac{1}{2} \mathrm{mv}_{\mathrm{e}}^{2}=\frac{\mathrm{GMm}}{\mathrm{R}} ; \mathrm{v}_{\mathrm{e}}=\sqrt{\frac{\mathrm{GMm}}{\mathrm{R}}}=\sqrt{2 \mathrm{gR}}$
<br/> <br/>In tunnel body will perform SHM at centre $\mathrm{V}_{\mathrm{max}}=\mathrm{A\omega}$ (see chapter on SHM)
<br/> <br/>$=\frac{\mathrm{R} 2 \pi}{2 \pi \sqrt{\mathrm{R} / \mathrm{g}}}=\sqrt{\mathrm{gR}}=\frac{\mathrm{v}_{\mathrm{e}}}{\sqrt{2}}$</div>


tags: ['Physics', 'Gravitation', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
title:
the-escape-velocity-from-the-surface-of-earth-is-v-e-a-body-is-projected-with-velocity-3-v-e-with-what-constant-velocity-will-it-move-in-the-inter-pla

slug:
the-escape-velocity-from-the-surface-of-earth-is-v-e-a-body-is-projected-with-velocity-3-v-e-with-what-constant-velocity-will-it-move-in-the-inter-pla-28517

question:
<div class="question">The escape velocity from the surface of earth is <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>e</mi></msub><mo>.</mo></math> A body is projected with velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>v</mi><mi>e</mi></msub><mo>.</mo></math> With what constant velocity will it move in the inter planetary space?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mi>e</mi></msub></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>v</mi><mi>e</mi></msub></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt><msub><mi>v</mi><mi>e</mi></msub></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mn>2</mn></msqrt><msub><mi>v</mi><mi>e</mi></msub></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mn>2</mn></msqrt><msub><mi>v</mi><mi>e</mi></msub></math></span> </div>

answer:
<div class="solution"><p> If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> be the velocity in the inter planetary space and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mrow><mi>i</mi><mo>,</mo></mrow></msub><mo> </mo><msub><mi>U</mi><mi>f</mi></msub></math> initial and final PE</p><p>Then, by the law of conservation of energy,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi>i</mi></msub><mo>+</mo><msub><mi>K</mi><mi>i</mi></msub><mo>=</mo><msub><mi>U</mi><mi>f</mi></msub><mo>+</mo><msub><mi>K</mi><mi>f</mi></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mi>G</mi><msub><mi>M</mi><mi>e</mi></msub><mi>m</mi></mrow><msub><mi>R</mi><mi>e</mi></msub></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mfenced><mrow><mn>3</mn><msub><mi>v</mi><mi>e</mi></msub></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>-</mo><mi>G</mi><msub><mi>M</mi><mi>e</mi></msub></mrow><msub><mi>R</mi><mi>e</mi></msub></mfrac><mo>+</mo><mfrac><mn>9</mn><mn>2</mn></mfrac><msubsup><mi>v</mi><mi>e</mi><mn>2</mn></msubsup><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>v</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>  </mo><mfrac><mrow><mo>-</mo><mn>2</mn><mi>G</mi><msub><mi>M</mi><mi>e</mi></msub></mrow><msub><mi>R</mi><mi>e</mi></msub></mfrac><mo>+</mo><mn>9</mn><msubsup><mi>V</mi><mi>e</mi><mn>2</mn></msubsup><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>  </mo><mfrac><mrow><mo>-</mo><mn>2</mn><mi>g</mi><msubsup><mi>R</mi><mi>e</mi><mn>2</mn></msubsup></mrow><msub><mi>R</mi><mi>e</mi></msub></mfrac><mo>+</mo><mn>9</mn><msubsup><mi>v</mi><mi>e</mi><mn>2</mn></msubsup><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><mo>  </mo><mfenced close="]" open="["><mrow><mo>∵</mo><mi>G</mi><msub><mi>M</mi><mi>e</mi></msub><mo>=</mo><mi>g</mi><msubsup><mi>R</mi><mi>e</mi><mn>2</mn></msubsup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mi>g</mi><msub><mi>R</mi><mi>e</mi></msub><mo>+</mo><mn>9</mn><mo>×</mo><mn>2</mn><mi>g</mi><msub><mi>R</mi><mi>e</mi></msub><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced close="]" open="["><mrow><mo>∵</mo><msubsup><mi>v</mi><mi>e</mi><mn>2</mn></msubsup><mo>=</mo><mn>2</mn><mi>g</mi><msub><mi>R</mi><mi>e</mi></msub></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn><mo> </mo><mi>g</mi><msub><mi>R</mi><mi mathvariant="italic">e</mi></msub><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>  </mo><mi>v</mi><mo>=</mo><msqrt><mn>8</mn></msqrt><mo>·</mo><msqrt><mn>2</mn><mi>g</mi><msub><mi>R</mi><mi>e</mi></msub></msqrt><mo>=</mo><mn>2</mn><msqrt><mn>2</mn></msqrt><msub><mi>v</mi><mi>e</mi></msub></math></p></div>


tags: ['Mathematics', 'Definite Integration', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P2.db
title:
the-expression-0-n-x-d-x-0-n-x-d-x-where-x-and-x-are-integral-and-fractional-part-of-x-and-n-n-is-equal-to

slug:
the-expression-0-n-x-d-x-0-n-x-d-x-where-x-and-x-are-integral-and-fractional-part-of-x-and-n-n-is-equal-to-31252

question:
<div class="question">The expression <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>n</mi></msubsup><mfenced close="]" open="["><mi>x</mi></mfenced><mi>d</mi><mi>x</mi></mrow><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>n</mi></msubsup><mfenced close="}" open="{"><mi>x</mi></mfenced><mi>d</mi><mi>x</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mi>x</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mi>x</mi></mfenced></math> are integral and fractional part of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi>N</mi></math>, is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>n</mi></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>-</mo><mn>1</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>-</mo><mn>1</mn></math></span> </div>

answer:
<div class="solution"><p>We have, <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mi>n</mi></msubsup><mfenced close="]" open="["><mi>x</mi></mfenced><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mn>0</mn><mi>d</mi><mi>x</mi><mo>+</mo><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><mn>1</mn><mi>d</mi><mi>x</mi><mo>+</mo><msubsup><mo>∫</mo><mn>2</mn><mn>3</mn></msubsup><mn>2</mn><mi>d</mi><mi>x</mi><mo>+</mo><mo>…</mo><mo>+</mo><msubsup><mo>∫</mo><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mfenced><mrow><mn>2</mn><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>2</mn><mfenced><mrow><mn>3</mn><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mn>3</mn><mfenced><mrow><mn>4</mn><mo>-</mo><mn>3</mn></mrow></mfenced><mo>+</mo><mo>…</mo><mo>…</mo><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced close="}" open="{"><mrow><mi>n</mi><mo>-</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>3</mn><mo>+</mo><mo>…</mo><mo>.</mo><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mi>n</mi></msubsup><mfenced close="}" open="{"><mi>x</mi></mfenced><mi>d</mi><mi>x</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>n</mi></msubsup><mfenced><mrow><mi>x</mi><mo>-</mo><mfenced close="]" open="["><mi>x</mi></mfenced></mrow></mfenced><mi>d</mi><mi>x</mi><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfrac><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>n</mi></msubsup><mfenced close="]" open="["><mi>x</mi></mfenced><mi>d</mi><mi>x</mi></mrow><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>n</mi></msubsup><mfenced close="}" open="{"><mi>x</mi></mfenced><mi>d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mi>n</mi><mo>-</mo><mn>1</mn></math></p></div>


title: the-false-statement-among-the-followings
slug: the-false-statement-among-the-followings-60438
question: <div class="question">The false statement among the followings:</div>
tags: ['Chemistry', 'Environmental Chemistry', 'BITSAT', 'BITSAT 2009']
answer: <div class="solution">The average residence time of NO is 4 days.</div>
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">The average residence time of $\mathrm{NO}$ is one month.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Limestone acts as a sink for $\mathrm{SO}_{\mathrm{x}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{SO}_{\mathrm{x}}$ can be removed from flue gases by passing through a solution of citrate ions.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Ammonia acts as a sink for $\mathrm{NO}_{\mathrm{x}^{-}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The average residence time of $\mathrm{NO}$ is one month.</span> </div>


title: the-family-of-straight-lines-2-a3-b-xab-y2-a4-b0-is-concurrent-at-the-point
slug: the-family-of-straight-lines-2-a3-b-xab-y2-a4-b0-is-concurrent-at-the-point-81029
tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P2.db
question:
<div class="question">The family of straight lines \((2 a+3 b) x+(a-b) y+2 a-4 b=0\) is concurrent at the point</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\left(\frac{2}{5}, \frac{-14}{5}\right)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\left(\frac{-2}{5}, \frac{-14}{5}\right)\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\left(\frac{-2}{5}, \frac{14}{5}\right)\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\left(\frac{2}{5}, \frac{14}{5}\right)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\left(\frac{2}{5}, \frac{-14}{5}\right)\)</span> </div>

answer:
<div class="solution">Rewriting the equation \((2 x+y+2) a+(3 x-y-4) b=0\) and for all \(a\), \(\mathrm{b}\) the straight lines pass through the intersection of \(2 x+y+2=0\) and \(3 x-y-4=0\) i.e., the point \(\left(\frac{2}{5},-\frac{14}{5}\right)\).</div>


tags: ['Physics', 'Thermodynamics', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
the-figure-shows-the-graph-of-p-v-t-versus-p-for-2-10-4-kg-of-hydrogen-gas-at-two-different-temperatures-where-p-v-and-t-represents-pressure-volume-an-1

slug:
the-figure-shows-the-graph-of-p-v-t-versus-p-for-2-10-4-kg-of-hydrogen-gas-at-two-different-temperatures-where-p-v-and-t-represents-pressure-volume-an-1-27970

question:
<div class="question"><p>The figure shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>p</mi><mi>V</mi></mrow><mi>T</mi></mfrac></math> versus <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup><mo> </mo><mo> </mo><mi>kg</mi></math> of hydrogen gas at two different temperatures, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>V</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> represents pressure, volume and temperature respectively.Then, the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>p</mi><mi>V</mi></mrow><mi>T</mi></mfrac><mo>,</mo></math> where the curve meet on the vertical axis, is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/12c359dd-c6a7-4166-a2e5-aff6c707f3e3-image.png" style="width: 220px; height: 182px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>R</mi><mn>5</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo> </mo><mi mathvariant="normal">R</mi></mrow><mn>3</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>R</mi><mn>10</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>R</mi><mn>15</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>R</mi><mn>10</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>Number of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>moles</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup><mo> </mo><mi>kg</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mfrac><mrow><mo> </mo><mi>Mass</mi><mo> </mo><mi>of</mi><mo> </mo><mi>hydrogen</mi><mo> </mo><mi>in</mi><mo> </mo><mi>gram</mi><mo> </mo></mrow><mrow><mo> </mo><mi>Molecul</mi><mi>a</mi><mi>r</mi><mo> </mo><mi>mass</mi><mo> </mo></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup></mrow><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo> </mo><mi>mole</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo>  </mo><mi>p</mi><mi>V</mi><mo>=</mo><mi>μ</mi><mi>R</mi><mi>T</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>p</mi><mi>V</mi></mrow><mi>T</mi></mfrac><mo>=</mo><mi>μ</mi><mi>R</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>×</mo><mi>R</mi><mo>=</mo><mfrac><mi>R</mi><mn>10</mn></mfrac></math></p></div>


tags: ['Physics', 'Thermodynamics', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
title:
the-figure-shows-the-graph-of-p-v-t-versus-p-for-2-10-4-kg-of-hydrogen-gas-at-two-different-temperatures-where-p-v-and-t-represents-pressure-volume-an

slug:
the-figure-shows-the-graph-of-p-v-t-versus-p-for-2-10-4-kg-of-hydrogen-gas-at-two-different-temperatures-where-p-v-and-t-represents-pressure-volume-an-15029

question:
<div class="question"><p>The figure shows the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>p</mi><mi>V</mi></mrow><mi>T</mi></mfrac></math> versus <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup><mo> </mo><mo> </mo><mi>kg</mi></math> of hydrogen gas at two different temperatures, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>V</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> represents pressure, volume and temperature respectively.Then, the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>p</mi><mi>V</mi></mrow><mi>T</mi></mfrac><mo>,</mo></math> where the curve meet on the vertical axis, is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/12c359dd-c6a7-4166-a2e5-aff6c707f3e3-image.png" style="width: 220px; height: 182px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>R</mi><mn>5</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mo> </mo><mi mathvariant="normal">R</mi></mrow><mn>3</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>R</mi><mn>10</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>R</mi><mn>15</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>R</mi><mn>10</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>Number of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>moles</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup><mo> </mo><mi>kg</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mfrac><mrow><mo> </mo><mi>Mass</mi><mo> </mo><mi>of</mi><mo> </mo><mi>hydrogen</mi><mo> </mo><mi>in</mi><mo> </mo><mi>gram</mi><mo> </mo></mrow><mrow><mo> </mo><mi>Molecul</mi><mi>a</mi><mi>r</mi><mo> </mo><mi>mass</mi><mo> </mo></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>4</mn></mrow></msup><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup></mrow><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo> </mo><mi>mole</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo>  </mo><mi>p</mi><mi>V</mi><mo>=</mo><mi>μ</mi><mi>R</mi><mi>T</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>p</mi><mi>V</mi></mrow><mi>T</mi></mfrac><mo>=</mo><mi>μ</mi><mi>R</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>×</mo><mi>R</mi><mo>=</mo><mfrac><mi>R</mi><mn>10</mn></mfrac></math></p></div>


title: the-final-product-obtained-in-the-reaction-1
slug: the-final-product-obtained-in-the-reaction-1-74331
tags: ['Chemistry', 'Hydrocarbons', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
question:
<div class="question">The final product obtained in the reaction
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/tgE9NbVlSaBeSvMpora0LZNAckjdKx1TSwilkR0ackA.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/wpGveGUr2ksq3Vxy4qAiB7svtxDeCJ4NQkEu44KlOFY.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/W0jzY_jNigfZHfRs3Q-2u12uTKh-VEkHuHBMa_n8aXk.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/IG8DBF5N2k_EgioLmFwV9ggvzOQQBHJq9CnTWympqTI.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/lWAZtiRWvJmYZAcNJBzSu6Wd55A8Xtsbki8oaFFSkBo.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/IG8DBF5N2k_EgioLmFwV9ggvzOQQBHJq9CnTWympqTI.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/onbBA_bOEyM_Xf4WalBQ-2V5Q8Bo7fXvf1VTourZMgc.original.fullsize.png"/></div>


tags: ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
title:
the-final-step-for-extraction-of-copper-from-copper-pyrite-in-bessemer-converter-involves-the-reaction

slug:
the-final-step-for-extraction-of-copper-from-copper-pyrite-in-bessemer-converter-involves-the-reaction-50915

question:
<div class="question">The final step for extraction of copper from copper pyrite in bessemer converter involves the reaction</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Cu</mi><mn>2</mn></msub><mo> </mo><mi mathvariant="normal">S</mi><mo>+</mo><mn>2</mn><msub><mi>Cu</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo>⟶</mo><mn>6</mn><mi>Cu</mi><mo>+</mo><msub><mi>SO</mi><mn>2</mn></msub></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msub><mi>Cu</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo>+</mo><mi>FeS</mi><mo>⟶</mo><mn>8</mn><mi>Cu</mi><mo>+</mo><msub><mi>FeSO</mi><mn>4</mn></msub></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msub><mi>Cu</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo>+</mo><mi>FeS</mi><mo>⟶</mo><mn>4</mn><mi>Cu</mi><mo>+</mo><mi>Fe</mi><mo>+</mo><msub><mi>SO</mi><mn>2</mn></msub></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Cu</mi><mn>2</mn></msub><mo> </mo><mi mathvariant="normal">S</mi><mo>+</mo><mn>2</mn><mi>FeO</mi><mo>⟶</mo><mn>2</mn><mi>Cu</mi><mo>+</mo><mi>Fe</mi><mo>+</mo><msub><mi>SO</mi><mn>2</mn></msub></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Cu</mi><mn>2</mn></msub><mo> </mo><mi mathvariant="normal">S</mi><mo>+</mo><mn>2</mn><msub><mi>Cu</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo>⟶</mo><mn>6</mn><mi>Cu</mi><mo>+</mo><msub><mi>SO</mi><mn>2</mn></msub></math></span> </div>

answer:
<div class="solution"><p>In Bessemer converter, copper-sulphide is partially oxidised to cuprous oxide, which further reacts with the remaining copper sulphide to give copper and sulphur di oxide, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Cu</mi><mn>2</mn></msub><mo> </mo><mi mathvariant="normal">S</mi><mo>+</mo><mn>2</mn><msub><mi>Cu</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo>⟶</mo><mn>6</mn><mi>Cu</mi><mo>+</mo><msub><mi>SO</mi><mn>2</mn></msub></math></p><p>This reaction represents auto reduction. <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Cu</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi></math> reacts with <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Cu</mi><mn>2</mn></msub><mi mathvariant="normal">S</mi></math> to give copper metal. The solid metal obtained is called as blister copper and is about 99% pure.</p></div>


title: the-first-emission-line-in-the-atomic-spectrum-of-hydrogen-in-the-balmer-series-appears-at
slug: the-first-emission-line-in-the-atomic-spectrum-of-hydrogen-in-the-balmer-series-appears-at-57423
tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
question:
<div class="question">The first emission line in the atomic spectrum of hydrogen in the Balmer series appears at</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{9 R}{400} \mathrm{~cm}^{-1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{7 \mathrm{R}}{144} \mathrm{~cm}^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3 \mathrm{R}}{4} \mathrm{~cm}^{-1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{5 \mathrm{R}}{36} \mathrm{~cm}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5 \mathrm{R}}{36} \mathrm{~cm}^{-1}$</span> </div>

answer:
<div class="solution">For Balmer $\mathrm{n}_{1}=2$ and $\mathrm{n}_{2}=3$; $v=R\left(\frac{1}{2^{2}}-\frac{1}{3^{2}}\right)=\frac{5 R}{36} \mathrm{~cm}^{-1}$</div>


tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P2.db
title:
the-focal-length-of-a-thin-convex-lens-for-red-and-blue-rays-are-100-cm-and-96-8-cm-respectively-then-the-dispersive-power-of-the-material-of-the-lens

slug:
the-focal-length-of-a-thin-convex-lens-for-red-and-blue-rays-are-100-cm-and-96-8-cm-respectively-then-the-dispersive-power-of-the-material-of-the-lens-88295

question:
<div class="question">The focal length of a thin convex lens for red and blue rays are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mi>cm</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>96</mn><mo>.</mo><mn>8</mn><mo> </mo><mi>cm</mi></math> respectively. Then, the dispersive power of the material of the lens is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>968</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>98</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0325</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>325</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0325</mn></math></span> </div>

answer:
<div class="solution"><blockquote><p>Dispersive power is ratio of angular dispersion and mean deviation.</p><p>It can be written in terms of focal length as,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mfrac><mrow><msub><mi>f</mi><mi>r</mi></msub><mo>-</mo><msub><mi>f</mi><mi>b</mi></msub></mrow><msub><mi>f</mi><mi>y</mi></msub></mfrac></math></p><p>Here <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>r</mi></msub><mo>=</mo><mn>100</mn><mo> </mo><mi>cm</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>b</mi></msub><mo>=</mo><mn>96</mn><mo>.</mo><mn>8</mn><mo> </mo><mi>cm</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>y</mi></msub><mo>=</mo><msqrt><msub><mi>f</mi><mi>b</mi></msub><mo>×</mo><msub><mi>f</mi><mi>r</mi></msub></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mn>96</mn><mo>.</mo><mn>8</mn><mo>×</mo><mn>100</mn></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≈</mo><mn>98</mn><mo>.</mo><mn>34</mn><mo> </mo><mi>cm</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Dispersive power</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ω</mi><mo>=</mo><mfrac><mrow><msub><mi>f</mi><mi>r</mi></msub><mo>-</mo><msub><mi>f</mi><mi>b</mi></msub></mrow><msub><mi>f</mi><mi>y</mi></msub></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>100</mn><mo>-</mo><mn>96</mn><mo>.</mo><mn>8</mn></mrow><mrow><mn>98</mn><mo>.</mo><mn>34</mn></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≈</mo><mn>0</mn><mo>.</mo><mn>0325</mn></math><br/>Hence, the dispersive power of the material of the lens is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0325</mn></math>.</p></blockquote></div>


title: the-following-change-can-be-carried-out-with
slug: the-following-change-can-be-carried-out-with-30983
tags: ['Chemistry', 'Aldehydes and Ketones', 'BITSAT', 'BITSAT 2020']
answer: <div class="solution">$\mathrm{LiAlH}_{4}$ will give the desired compound.</div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The following change can be carried out with
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/JuYZjS8coSEKWn92PSFrnPvWWetcLYOiQMTONFWbSX4.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{NaBH}_{4}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{LiAlH}_{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{H}_{2} / \mathrm{Pt}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{PCC} / \mathrm{CH}_{2} \mathrm{Cl}_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{LiAlH}_{4}$</span> </div>


tags: ['Chemistry', 'Chemical Kinetics', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">One</span> </div>
source_db: MarksBatch2_P2.db
title:
the-following-data-are-for-the-decomposition-of-ammonium-nitrite-in-aqueous-solution-the-order-of-reaction-is-1

slug:
the-following-data-are-for-the-decomposition-of-ammonium-nitrite-in-aqueous-solution-the-order-of-reaction-is-1-80864

question:
<div class="question">The following data are for the decomposition of ammonium nitrite in aqueous solution :<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/C2gSp2yRHw6YU0u2ydT16ZtNdhOkIxS7eFDlTyw7eaI.original.fullsize.png"/><br/><br/>The order of reaction is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Zero</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">One</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Two</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Three</span> </li> </ul>

answer:
<div class="solution">$\mathrm{NH}_{4} \mathrm{NO}_{2} \longrightarrow \mathrm{N}_{2}+2 \mathrm{H}_{2} \mathrm{O}$<br/>Volume of $\mathrm{N}_{2}$ formed in successive five minutes are $2.75 \mathrm{cc}, 2.40 \mathrm{cc}$ and $2.25 \mathrm{cc}$ which is in decreasing order. So rate of reaction is dependent on concentration of $\mathrm{NH}_{4} \mathrm{NO}_{2}$. As decrease is not very fast so it will be first order reaction.</div>


tags: ['Chemistry', 'Chemical Kinetics', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">One</span> </div>
source_db: MarksBatch2_P2.db
title:
the-following-data-are-for-the-decomposition-of-ammonium-nitrite-in-aqueous-solution-the-order-of-reaction-is

slug:
the-following-data-are-for-the-decomposition-of-ammonium-nitrite-in-aqueous-solution-the-order-of-reaction-is-29240

question:
<div class="question">The following data are for the decomposition of ammonium nitrite in aqueous solution :<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/C2gSp2yRHw6YU0u2ydT16ZtNdhOkIxS7eFDlTyw7eaI.original.fullsize.png"/><br/><br/>The order of reaction is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Zero</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">One</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Two</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Three</span> </li> </ul>

answer:
<div class="solution">$\mathrm{NH}_{4} \mathrm{NO}_{2} \longrightarrow \mathrm{N}_{2}+2 \mathrm{H}_{2} \mathrm{O}$<br/>Volume of $\mathrm{N}_{2}$ formed in successive five minutes are $2.75 \mathrm{cc}, 2.40 \mathrm{cc}$ and $2.25 \mathrm{cc}$ which is in decreasing order. So rate of reaction is dependent on concentration of $\mathrm{NH}_{4} \mathrm{NO}_{2}$. As decrease is not very fast so it will be first order reaction.</div>


title: the-following-reaction-is-known-as-2
slug: the-following-reaction-is-known-as-2-45438
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
question:
<div class="question">The following reaction is known as
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/n-Y_LjspTwG3TKHAJstC7iydoaPcjSNmAibfPHjWQKE.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Friedel-Craft's reaction</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Kolbe reaction</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Reimer-Tiemann reaction</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Witting reaction</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Kolbe reaction</span> </div>

answer:
<div class="solution">At $120-140^{\circ} \mathrm{C}$ temperature and 1.5 atm pressure, sodium phenoxide reacts with $\mathrm{CO}_{2}$ to yield sodium salicylate which on further hydrolysis give to salicylic acid. This reaction is known as Kolbe's reaction.</div>


title: the-following-two-compounds-are-1
slug: the-following-two-compounds-are-1-65957
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
question:
<div class="question">The following two compounds are
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/j-DZw5_jEzsKI1y7cBn9sOKe5SEWT6veeOWO4wN3298.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">enantiomers</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">diastereomers</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">identical</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">epimers</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">enantiomers</span> </div>

answer:
<div class="solution">The given two structures are optical isomers but as these are mirror image of each other, hence they represent enantiomers of each other.</div>


title: the-foot-of-the-perpendicular-from-the-point-7145-to-the-plane-2-x-4-y-z-2-are
slug: the-foot-of-the-perpendicular-from-the-point-7145-to-the-plane-2-x-4-y-z-2-are-92974
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
question:
<div class="question">The foot of the perpendicular from the point (7,14,5) to the plane $2 \mathrm{x}+4 \mathrm{y}-\mathrm{z}=2$ are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">(1,2,8)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">(3,2,8)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">(5,10,6)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">(9,18,4)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">(1,2,8)</span> </div>

answer:
<div class="solution">We know that the length of the perpendicular from the point $\left(x_{1}, y_{1}, z_{1}\right)$ to the plane $a x+b y+c z+d=0$ is $\frac{\left|a x_{1}+b y_{1}+c z_{1}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$
<br/>and the co-ordinate $(\alpha, \beta, \gamma)$ of the foot of the $\perp$ are given by $\frac{\alpha-x_{1}}{a}=\frac{\beta-y_{1}}{b}=\frac{\gamma-z_{1}}{c}$
<br/>$=-\left(\frac{a x_{1}+b y_{1}+c z_{1}+d}{a^{2}+b^{2}+c^{2}}\right)$ ...(1)
<br/>In the given ques, $x_{1}=7, y_{1}=14, z_{1}=5$,
<br/>$a=2 b=4, c=-1, d=-2$
<br/>By putting these values in (1), we get $\frac{\alpha-7}{2}=\frac{\beta-14}{4}=\frac{\gamma-5}{-1}=-\frac{63}{21}$
<br/>$\Rightarrow \alpha=1, \beta=2$ and $\gamma=8$
<br/>Hence, foot of $\perp$ is (1,2,8)</div>


tags: ['Physics', 'Electrostatics', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P2.db
title:
the-force-between-two-small-charged-spheres-having-charges-of-1-1-0-7-c-and-2-1-0-7-c-placed-20-cm-apart-in-air-is

slug:
the-force-between-two-small-charged-spheres-having-charges-of-1-1-0-7-c-and-2-1-0-7-c-placed-20-cm-apart-in-air-is-22532

question:
<div class="question">The force between two small charged spheres having charges of $1 \times 10^{-7} \mathrm{C}$ and $2 \times 10^{-7} \mathrm{C}$ placed $20 \mathrm{~cm}$ apart in air is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4.5 \times 10^{-2} \mathrm{~N}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5.4 \times 10^{-2} \mathrm{~N}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$4.5 \times 10^{-3} \mathrm{~N}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$5.4 \times 10^{-3} \mathrm{~N}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4.5 \times 10^{-3} \mathrm{~N}$</span> </div>

answer:
<div class="solution">Here, $q_1=1 \times 10^{-7} \mathrm{C}, q_2$ and $2 \times 10^{-7} \mathrm{C}$, $r=20 \mathrm{~cm}=20 \times 10^{-2} \mathrm{~m}$<br/>$F=\frac{q_1 q_2}{4 \pi \varepsilon_0 r^2}=\frac{9 \times 10^9 \times 1 \times 10^{-7} \times 2 \times 10^{-7}}{\left(20 \times 10^{-2}\right)^2}$<br/>$=4.5 \times 10^{-3} N$</div>


tags: ['Physics', 'Laws of Motion', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
title:
the-force-on-a-body-of-mass-1-kg-is-20-i-10-j-n-if-it-starts-from-rest-then-the-position-of-the-body-at-time-t-2-s-is

slug:
the-force-on-a-body-of-mass-1-kg-is-20-i-10-j-n-if-it-starts-from-rest-then-the-position-of-the-body-at-time-t-2-s-is-49728

question:
<div class="question">The force on a body of mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mi>kg</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>20</mn><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>10</mn><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></mrow></mfenced><mo> </mo><mi mathvariant="normal">N</mi></math>. If it starts from rest, then the position of the body at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn><mo> </mo><mi mathvariant="normal">s</mi><mo>,</mo></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>20</mn><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mn>40</mn><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mn>40</mn><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>-</mo><mn>20</mn><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>20</mn><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>20</mn><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mn>20</mn><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>10</mn><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>x</mi></msub><mo>=</mo><mn>20</mn><mo> </mo><mi mathvariant="normal">N</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>y</mi></msub><mo>=</mo><mn>10</mn><mo> </mo><mi mathvariant="normal">N</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Acceleration,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>x</mi></msub><mo>=</mo><mfrac><msub><mi>F</mi><mi>x</mi></msub><mi>m</mi></mfrac><mo>=</mo><mfrac><mn>20</mn><mn>1</mn></mfrac></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>x</mi></msub><mo>=</mo><mn>20</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mi>y</mi></msub><mo>=</mo><mfrac><msub><mi>F</mi><mi>y</mi></msub><mi>m</mi></mfrac><mo>=</mo><mfrac><mn>10</mn><mn>1</mn></mfrac><mo>=</mo><mn>10</mn><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Displacement of body in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> -direction in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">s</mi></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msub><mi>u</mi><mi>x</mi></msub><mo> </mo><mi>t</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>a</mi><mi>x</mi></msub><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>20</mn><mo>×</mo><msup><mn>2</mn><mn>2</mn></msup></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>40</mn><mo> </mo><mi mathvariant="normal">m</mi></math></p><p>Displacement of body in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-direction in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">s</mi></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>u</mi><mi>y</mi></msub><mo> </mo><mi>t</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo> </mo><msub><mi>a</mi><mi>y</mi></msub><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>10</mn><mo>×</mo><msup><mn>2</mn><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>20</mn><mo> </mo><mi mathvariant="normal">m</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Position of the body at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn><mo> </mo><mi mathvariant="normal">s</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mi>x</mi><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mi>y</mi><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>40</mn><mo> </mo><mover><mi mathvariant="normal">i</mi><mo>^</mo></mover><mo>+</mo><mn>20</mn><mo> </mo><mover><mi mathvariant="normal">j</mi><mo>^</mo></mover></math></p></div>


title: the-formation-of-co-and-co-2-illustrates-the-law-of
slug: the-formation-of-co-and-co-2-illustrates-the-law-of-33827
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P2.db
question:
<div class="question">The formation of $\mathrm{CO}$ and $\mathrm{CO}_{2}$ illustrates the law of</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">reciprocal proportion</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">conservation of mass</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">multiple proportion</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">constant composition</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">multiple proportion</span> </div>

answer:
<div class="solution">Formation of $\mathrm{CO}$ and $\mathrm{CO}_{2}$ illustrates the law of multiple proportion that is constant mass of $\mathrm{C}$ reacts with different masses of oxygen. These masses here bears simple ratio of 1: 2 .</div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P2.db
title:
the-fourth-term-of-an-ap-is-three-times-of-the-first-term-and-the-seventh-term-exceeds-the-twice-of-the-third-term-by-one-then-the-common-difference-o

slug:
the-fourth-term-of-an-ap-is-three-times-of-the-first-term-and-the-seventh-term-exceeds-the-twice-of-the-third-term-by-one-then-the-common-difference-o-12364

question:
<div class="question">The fourth term of an A.P. is three times of the first term and the seventh term exceeds the twice of the third term by one, then the common difference of the progression is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-1</span> </li> </ul>

answer:
<div class="solution">Let the progression be $a$, $a+d$, a +2 d,
<br/> <br/>Then $x_{4}=3 x_{1} \Rightarrow a+3 d=3 a \Rightarrow 3 d=2 a \ldots$ (i)
<br/> <br/>Again $x_{7}=2 x_{3}+1$
<br/> <br/>$\Rightarrow a+6 d=2(a+2 d)+1 \Rightarrow 2 d=a+1 \quad \ldots$ (ii)
<br/> <br/>Solving (i) and (ii), we get $\mathrm{a}=3, \mathrm{~d}=2$</div>


title: the-fraction-of-total-volume-occupied-by-the-atoms-present-in-a-simple-cube-is
slug: the-fraction-of-total-volume-occupied-by-the-atoms-present-in-a-simple-cube-is-15975
tags: ['Chemistry', 'Solid State', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P2.db
question:
<div class="question">The fraction of total volume occupied by the atoms present in a simple cube is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{3 \sqrt{2}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{4 \sqrt{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{4}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{6}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\pi}{6}$</span> </div>

answer:
<div class="solution">Number of atoms per unit cell $=1$ Atoms touch each other along edges. Hence $r=\frac{a}{2}$<br/>$(\mathrm{r}=$ radius of atom and $\mathrm{a}=$ edge length)<br/>Therefore $\%$ $=\frac{\frac{4}{3} \pi r^{3}}{(2 r)^{3}}=\frac{\pi}{6}$</div>


tags: ['Physics', 'Magnetic Properties of Matter', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2 \%$</span> </div>
source_db: MarksBatch2_P2.db
title:
the-free-space-inside-a-current-carrying-toroid-is-filled-with-a-material-of-susceptibility-2-1-0-2-the-percentage-increase-in-the-value-of-magnetic-f

slug:
the-free-space-inside-a-current-carrying-toroid-is-filled-with-a-material-of-susceptibility-2-1-0-2-the-percentage-increase-in-the-value-of-magnetic-f-19243

question:
<div class="question">The free space inside a current carrying toroid is filled with a material of susceptibility $2 \times 10^{-2}$. The percentage increase in the value of magnetic field inside the toroid will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.2 \%$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.1 \%$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> $1 \%$</span> </li> </ul>

answer:
<div class="solution">Given,<br/>Susceptibility of material, $\chi_m=2 \times 10^{-2}$<br/>Using $\mu_{\mathrm{r}}=1+\chi_{\mathrm{m}}=1+0.02=1.02$<br/>$\mathrm{B}_{\text {final }}=\mu_{\mathrm{r}} \mathrm{B}_0$ (here, $\mathrm{B}_0=$ initial magnetic field)<br/>$\%$ increase in magnetic field<br/>$=\frac{\mathrm{B}_{\text {final }}-\mathrm{B}_0}{\mathrm{~B}_0} \times 100=\frac{\mu_{\mathrm{r}} \mathrm{B}_0-\mathrm{B}_0 \times 100}{\mathrm{~B}_0}$<br/>$=\frac{(\chi+1)-1 \times 100}{1}=0.02 \times 100=2 \%$</div>


tags: ['Physics', 'Dual Nature of Matter', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
title:
the-frequency-and-the-intensity-of-incident-beam-of-light-falling-on-the-surface-of-a-photoelectric-material-is-increased-by-a-factor-of-two-this-will

slug:
the-frequency-and-the-intensity-of-incident-beam-of-light-falling-on-the-surface-of-a-photoelectric-material-is-increased-by-a-factor-of-two-this-will-33774

question:
<div class="question">The frequency and the intensity of incident beam of light falling on the surface of a photoelectric material is increased by a factor of two. This will</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">increase the maximum kinetic energy of the ejected photoelectrons by a factor of more than two and would increase the photoelectric current by a factor of two.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">increase the maximum kinetic energy of the photo electron and would increase the photoelectric current both by a factor of two.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">increase the maximum kinetic energy of the photoelectron by a factor of two and will have no effect on the magnitude of the photoelectric current produced.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">not produce any effect on the kinetic energy of the emitted photoelectrons but will increase the photoelectric current by a factor of two.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">increase the maximum kinetic energy of the ejected photoelectrons by a factor of more than two and would increase the photoelectric current by a factor of two.</span> </div>

answer:
<div class="solution"><p>Using Einstein's photoelectric equations,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mi>v</mi><mo>=</mo><mi>ϕ</mi><mo>+</mo><mi>e</mi><msub><mi>V</mi><mi>s</mi></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><msub><mi>V</mi><mi>s</mi></msub><mo>=</mo><mi>h</mi><mi>v</mi><mo>-</mo><mi>ϕ</mi><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi mathvariant="normal">i</mi><mo>)</mo></math></p><p>Thus, when the frequency of incident light radiation (photons) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> increases to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>v</mi></math>, the stopping potential <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>s</mi></msub></math> changes to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>s</mi></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><msubsup><mi>V</mi><mi>s</mi><mo>'</mo></msubsup><mo>=</mo><mi>h</mi><mo>(</mo><mn>2</mn><mi>v</mi><mo>)</mo><mo>-</mo><mi>ϕ</mi><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>ii</mi></mfenced></math></p><p>On dividing Eq. (ii) by Eq. (ii), we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><msub><mi>V</mi><mi>s</mi></msub><mo>'</mo></msup><msub><mi>V</mi><mi>s</mi></msub></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>h</mi><mi>v</mi><mo>-</mo><mi>ϕ</mi></mrow><mrow><mi>h</mi><mi>v</mi><mo>-</mo><mi>ϕ</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><mi>h</mi><mi>v</mi><mo>-</mo><mn>2</mn><mi>ϕ</mi><mo>+</mo><mi>ϕ</mi></mrow><mrow><mi>h</mi><mi>v</mi><mo>-</mo><mi>ϕ</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mo>(</mo><mi>h</mi><mi>v</mi><mo>-</mo><mi>ϕ</mi><mo>)</mo><mo>+</mo><mi>ϕ</mi></mrow><mrow><mi>h</mi><mi>v</mi><mo>-</mo><mi>ϕ</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mo>+</mo><mfrac><mi>ϕ</mi><mrow><mi>h</mi><mi>v</mi><mo>-</mo><mi>ϕ</mi></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfrac><msub><msup><mi>V</mi><mo>'</mo></msup><mi>s</mi></msub><msub><mi>V</mi><mi>s</mi></msub></mfrac><mo>&gt;</mo><mn>2</mn><mo>⇒</mo><msubsup><mi>V</mi><mi>s</mi><mo>'</mo></msubsup><mo>&gt;</mo><mn>2</mn><msub><mi>V</mi><mi>s</mi></msub></math></p></div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.77</span> </div>
source_db: MarksBatch2_P2.db
title:
the-frequency-of-a-sonometer-wire-is-100-hz-when-the-weights-producing-the-tensions-are-completely-immersed-in-water-the-frequency-becomes-80-hz-and-o

slug:
the-frequency-of-a-sonometer-wire-is-100-hz-when-the-weights-producing-the-tensions-are-completely-immersed-in-water-the-frequency-becomes-80-hz-and-o-68069

question:
<div class="question">The frequency of a sonometer wire is $100 \mathrm{~Hz}$. When the weights producing the tensions are completely immersed in water, the frequency becomes $80 \mathrm{~Hz}$ and on immersing the weights in a certain liquid, the frequency becomes $60 \mathrm{~Hz}$. The specific gravity of the liquid is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1.42</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">1.77</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1.82</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1.21</span> </li> </ul>

answer:
<div class="solution">As we know, frequency $f \propto \sqrt{m g}$ or $f \propto \sqrt{g}$
<br/> <br/>In water, $\mathrm{f}_{\mathrm{w}}=0.8 \mathrm{f}_{\text {air }}$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\frac{\mathrm{g}^{\prime}}{\mathrm{g}}(0.8)^{2}=0.64 \\
<br/> <br/>\Rightarrow 1-\frac{\rho_{\mathrm{w}}}{\rho_{\mathrm{m}}}=0.64 \\
<br/> <br/>\Rightarrow \frac{\rho_{\mathrm{w}}}{\rho_{\mathrm{m}}}=0.36
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>In liquid, $\frac{g^{\prime}}{g}=(0.6)^{2}=0.36$
<br/> <br/>$$
<br/> <br/>1-\frac{\rho_{1}}{\rho_{\mathrm{m}}}=0.36 \frac{\rho_{1}}{\rho_{\mathrm{m}}}=0.64
<br/> <br/>$$
<br/> <br/>From eq. (1) and (2) $\frac{\rho_{l}}{\rho_{n}}=\frac{0.64}{0.36} \therefore \rho_{l}=1.77$</div>


tags: ['Chemistry', 'Structure of Atom', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
title:
the-frequency-of-radiation-emitted-when-the-electron-falls-from-n-4-to-n-1-in-a-hydrogen-atom-will-be-given-ionization-energy-of-h-218-1-0-18-j-atom-1

slug:
the-frequency-of-radiation-emitted-when-the-electron-falls-from-n-4-to-n-1-in-a-hydrogen-atom-will-be-given-ionization-energy-of-h-218-1-0-18-j-atom-1-61046

question:
<div class="question">The frequency of radiation emitted when the electron falls from $\mathrm{n}=4$ to $\mathrm{n}=1$ in a hydrogen atom will be (Given : ionization energy of $\mathrm{H}=2.18 \times 10^{-18} \mathrm{~J}$ atom ${ }^{-1}$ and $\mathrm{h}=6.625 \times 10^{-34} \mathrm{Js}$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.54 \times 10^{15} \mathrm{~s}^{-1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1.03 \times 10^{15} \mathrm{~s}^{-1}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$3.08 \times 10^{15} \mathrm{~s}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.00 \times 10^{15} \mathrm{~s}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3.08 \times 10^{15} \mathrm{~s}^{-1}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \mathrm{v} &amp;=\frac{1}{\mathrm{~h}} \times \mathrm{IE} \times\left[\frac{1}{\mathrm{n}_{1}{ }^{2}}-\frac{1}{\mathrm{n}_{2}{ }^{2}}\right] \\ &amp;=\frac{2.18 \times 10^{-18}}{6.625 \times 10^{-34}} \times\left[\frac{1}{1}-\frac{1}{16}\right]=3.08 \times 10^{15} \mathrm{~s}^{-1} \end{aligned}$</div>


tags: ['Physics', 'Units and Dimensions', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
title:
the-frequency-of-vibration-of-string-is-given-by-v-2-l-p-m-f-12-here-p-is-number-of-segments-in-the-string-and-l-is-the-length-the-dimensional-formula

slug:
the-frequency-of-vibration-of-string-is-given-by-v-2-l-p-m-f-12-here-p-is-number-of-segments-in-the-string-and-l-is-the-length-the-dimensional-formula-57069

question:
<div class="question">The frequency of vibration of string is given by
<br/> <br/>$\mathrm{v}=\frac{\mathrm{p}}{2 l}\left[\frac{\mathrm{F}}{\mathrm{m}}\right]^{1 / 2}$
<br/> <br/>Here $\mathrm{p}$ is number of segments in the string and $l$ is the length. The dimensional formula for $\mathrm{m}$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{M}^{0} \mathrm{LT}^{-1}\right]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{ML}^{0} \mathrm{~T}^{-1}\right]$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{ML}^{-1} \mathrm{~T}^{0}\right]$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{ML}^{-1} \mathrm{~T}^{0}\right]$</span> </div>

answer:
<div class="solution">$\mathrm{v}=\frac{\mathrm{p}}{2 \ell}\left[\frac{\mathrm{F}}{\mathrm{m}}\right]^{1 / 2}$
<br/> <br/>$v^{2}=\frac{p}{4 \ell^{2}} \frac{F}{m} \Rightarrow m=\frac{p^{2} F}{4 \ell^{2} v^{2}}$
<br/> <br/>Now, dimensional formula of R.H.S.
<br/> <br/>$=\frac{\mathrm{MLT}^{-2}}{\mathrm{~L}^{2}\left(\frac{1}{\mathrm{~T}}\right)^{2}}$
<br/> <br/>[p will have no dimension as it is an integer] $=\mathrm{ML}^{-1} \mathrm{~T}^{0}$
<br/> <br/>So, dimensions of $\mathrm{m}$ will be $\mathrm{ML}^{-1} \mathrm{~T}^{0}$.</div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
title:
the-frequency-of-whistle-of-an-engine-appears-to-be-45-th-of-initial-frequency-when-it-crosses-a-stationary-observer-if-the-velocity-of-sound-is-330-m

slug:
the-frequency-of-whistle-of-an-engine-appears-to-be-45-th-of-initial-frequency-when-it-crosses-a-stationary-observer-if-the-velocity-of-sound-is-330-m-46101

question:
<div class="question">The frequency of whistle of an engine appears to be $(4 / 5)^{\text {th }}$ of initial frequency when it crosses a stationary observer. If the velocity of sound is $330 \mathrm{~m} / \mathrm{s}$, then the speed of engine will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$30 \mathrm{~m} / \mathrm{s}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$36.6 \mathrm{~m} / \mathrm{s}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$40 \mathrm{~m} / \mathrm{s}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$330 \mathrm{~m} / \mathrm{s}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$36.6 \mathrm{~m} / \mathrm{s}$</span> </div>

answer:
<div class="solution">$n^{\prime}=\frac{n v}{v-v_{s}} \ldots \ldots \ldots(1)$
<br/> <br/>$n^{\prime \prime}=\frac{n v}{v+v_{s}} \ldots \ldots . .(2)$
<br/> <br/>From (1) and (2), $\frac{n^{\prime}}{n^{\prime \prime}}=\frac{v+v_{s}}{v-v_{s}} \ldots \ldots . .(3)$
<br/> <br/>According to question, $\frac{n^{\prime}}{n^{\prime}}=\frac{5}{4}$
<br/> <br/>$v_{s}=? v=330 \mathrm{~m} / \mathrm{s} \ldots \ldots \ldots .(4)$
<br/> <br/>From eq. (3) and (4)
<br/> <br/>$\frac{5}{4}=\left[\frac{330+v_{s}}{330-v_{s}}\right] \Rightarrow 9 v_{s}=330$
<br/> <br/>$\therefore v_{s}=36.6 \mathrm{~m} / \mathrm{s}$</div>


tags: ['Physics', 'Units and Dimensions', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P2.db
title:
the-frequency-v-of-an-oscillating-liquid-drop-may-depend-upon-radius-r-of-the-drop-density-of-liquid-and-the-surface-tension-s-of-the-liquid-as-v-r-a-

slug:
the-frequency-v-of-an-oscillating-liquid-drop-may-depend-upon-radius-r-of-the-drop-density-of-liquid-and-the-surface-tension-s-of-the-liquid-as-v-r-a-99014

question:
<div class="question">The frequency $(v)$ of an oscillating liquid drop may depend upon radius $(r)$ of the drop, density $(\rho)$ of liquid and the surface tension $(s)$ of the liquid as :<br/>$v=r^a \rho^b s^c$. The values of $a, b$ and $c$ respectively are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(-\frac{3}{2},-\frac{1}{2} ; \frac{1}{2}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\frac{3}{2},-\frac{1}{2}, \frac{1}{2}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{3}{2}, \frac{1}{2} ;-\frac{1}{2}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(-\frac{3}{2}, \frac{1}{2}, \frac{1}{2}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(-\frac{3}{2},-\frac{1}{2} ; \frac{1}{2}\right)$</span> </div>

answer:
<div class="solution"><br/>$\begin{aligned} &amp; v=\mathrm{r}^{\mathrm{a}} \rho^{\mathrm{b}} \mathrm{s}^{\mathrm{c}} \Rightarrow[v]=[\mathrm{r}]^{\mathrm{a}}[\rho]^{\mathrm{b}}[\mathrm{s}]^{\mathrm{c}} \\ &amp; \Rightarrow \quad\left[\mathrm{T}^{-1}\right]=\left[\mathrm{L}^1\right]^{\mathrm{a}}\left[\mathrm{M}^1 \mathrm{~L}^{-3}\right]^{\mathrm{b}}\left[\frac{\mathrm{MLT}}{\mathrm{L}}\right]^{-2} \\ &amp; \Rightarrow \mathrm{T}^{-1}=\mathrm{M}^{\mathrm{b}+\mathrm{c}} \cdot \mathrm{L}^{\mathrm{a}-3 \mathrm{~b}} \cdot \mathrm{T}^{-2 c} \\ &amp; \Rightarrow \mathrm{b}+\mathrm{c}=0, \mathrm{a}-3 \mathrm{~b}=0,-2 \mathrm{c}=-1 \\ &amp; \Rightarrow \quad \mathrm{c}=\frac{1}{2}, \mathrm{~b}=-\frac{1}{2}, \mathrm{a}-3 \mathrm{~b}=0 ; \\ &amp; \mathrm{a}+\frac{3}{2}=0 \Rightarrow \mathrm{a}=-\frac{3}{2}\end{aligned}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">40</span> </div>
source_db: MarksBatch2_P2.db
title:
the-fuel-charges-for-running-a-train-are-proportional-to-the-square-of-the-speed-generated-in-miles-per-hour-and-costs-48-per-hour-at-16-miles-per-hou

slug:
the-fuel-charges-for-running-a-train-are-proportional-to-the-square-of-the-speed-generated-in-miles-per-hour-and-costs-48-per-hour-at-16-miles-per-hou-28972

question:
<div class="question">The fuel charges for running a train are proportional to the square of the speed generated in miles per hour and costs ' 48 per hour at 16 miles per hour. The most economical speed if the fixed charges i.e. salaries etc. amount to 300 per hour is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">20</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">30</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">40</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Let the speed of the train be $\mathrm{v}$ and distance to be covered be so that total time taken is $\mathrm{s} / \mathrm{v}$ hours. Cost of fuel per hour $=\mathrm{kv}^{2}(\mathrm{k}$ is constant) Also $48=\mathrm{k} .16^{2}$ by given condition $\therefore \mathrm{k}=\frac{3}{16}$
<br/> <br/>$\therefore$ Cost to fuel per hour $\frac{3}{16} \mathrm{v}^{2}$. Other charges per hour are 300 . Total running cost,
<br/> <br/>$$
<br/> <br/>\begin{array}{r}
<br/> <br/>C=\left(\frac{3}{16} \mathrm{v}^{2}+300\right) \frac{\mathrm{s}}{\mathrm{v}}=\frac{3 \mathrm{~s}}{16} \mathrm{v}+\frac{300 \mathrm{~s}}{\mathrm{v}} \\
<br/> <br/>\frac{\mathrm{dC}}{\mathrm{dv}}=\frac{3 \mathrm{~s}}{16}-\frac{300 \mathrm{~s}}{\mathrm{v}^{2}}=0 \Rightarrow \mathrm{v}=40
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>$\frac{\mathrm{d}^{2} \mathrm{C}}{\mathrm{dv}^{2}}=\frac{600 \mathrm{~s}}{\mathrm{v}^{3}}&gt;0 \therefore \mathrm{v}=40$ results in minimum
<br/> <br/>running cost</div>


title: the-function-f-x-2-x-x-2-has-a-local-minimum-at-1
slug: the-function-f-x-2-x-x-2-has-a-local-minimum-at-1-11970
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$x=1$</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The function $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{2}+\frac{2}{\mathrm{x}}$ has a local minimum at</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x=2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x=-2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x=0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$x=1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\frac{x}{2}+\frac{2}{x}$ is of the form $x+\frac{1}{x} \geq 2$ and equality holds for $x=1$</div>


title: the-function-f-x-lo-g-1-x-2-x-2-x-is-increasing-on-3
slug: the-function-f-x-lo-g-1-x-2-x-2-x-is-increasing-on-3-25213
question: <div class="question">The function $f(x)=\log (1+x)-\frac{2 x}{2+x}$ is increasing on</div>
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(0, \infty)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(-\infty, 0)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(-\infty, \infty)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(0, \infty)$</span> </div>

answer:
<div class="solution">Given $f(x)=\log (1+x)-\frac{2 x}{2+x}$<br/>$\begin{aligned}<br/>&amp; f^{\prime}(x)=\frac{1}{1+x}-\frac{(2+x)(2)-2 x}{(2+x)^2} \\<br/>&amp; =\frac{1}{1+x}-\frac{4}{(2+x)^2} \\<br/>&amp; =\frac{(2+x)^2-4-4 x}{(1+x)(2+x)^2}=\frac{x^2}{(1+x)(2+x)^2}&gt;0 \text { for all } \\<br/>&amp; x \in(0, \infty)<br/>\end{aligned}$<br/>Thus, given function $\mathrm{f}(\mathrm{x})$ is increasing on $(0$, $\infty)$.</div>


title: the-function-f-x-tan-1-sin-x-cos-x-is-an-increasing-function-in-1
slug: the-function-f-x-tan-1-sin-x-cos-x-is-an-increasing-function-in-1-24310
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">The function $f(x)=\tan ^{-1}(\sin x+\cos x)$ is an increasing function in</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left(-\frac{\pi}{2}, \frac{\pi}{4}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(0, \frac{\pi}{2}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(-\frac{\pi}{2}, \frac{\pi}{4}\right)$</span> </div>

answer:
<div class="solution">Since, $f(x)=\tan ^{-1}(\sin x+\cos x)$<br/>$\therefore \mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{1+(\sin \mathrm{x}+\cos \mathrm{x})^2}(\cos \mathrm{x}-\sin \mathrm{x})$<br/>$=\frac{\sqrt{2} \cos \left(x+\frac{\pi}{4}\right)}{1+(\sin x+\cos x)^2}$<br/>$f(x)$ is increasing if $f^{\prime}(x)&gt;0 \Rightarrow \cos \left(x+\frac{\pi}{4}\right)&gt;0$<br/>$\Rightarrow \quad-\frac{\pi}{2} &lt; x+\frac{\pi}{4} &lt; \frac{\pi}{2} \Rightarrow-\frac{3 \pi}{2} &lt; x+\frac{\pi}{4}$</div>


title: the-function-f-x-x-x-2-2-is-increasing-in-the-set
slug: the-function-f-x-x-x-2-2-is-increasing-in-the-set-40718
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P2.db
question:
<div class="question">The function $\mathrm{f}(\mathrm{x})=(\mathrm{x}(\mathrm{x}-2))^{2}$ is increasing in the set</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(-\infty, 0) \cup(2, \infty)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(-\infty, 1)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">(0,1)$\cup(2, \infty)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">(1,2)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">(0,1)$\cup(2, \infty)$</span> </div>

answer:
<div class="solution">$$
<br/> <br/>\text { } \begin{array}{l}
<br/> <br/>\text { Here, } \mathrm{f}(\mathrm{x})=(\mathrm{x}(\mathrm{x}-2))^{2} \\
<br/> <br/>\Rightarrow \mathrm{f}^{\prime}(\mathrm{x})=4 \mathrm{x}(\mathrm{x}-2)(\mathrm{x}-1)
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>For $\mathrm{f}(\mathrm{x})$ as increasing, $\mathrm{f}^{\prime}(\mathrm{x})&gt;0$
<br/> <br/>$$
<br/> <br/>\text { So, } 4 x(x-1)(x-2)&gt;0 \Rightarrow x(x-1)(x-2)&gt;0
<br/> <br/>$$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/lSKQqu4_r-O0TQCuFzGXYtr4R0q6W-FA-qe_tBAQ7TI.original.fullsize.png"/><br/> <br/> <br/>From the above figure required interval is, (0,1)$\cup(2, \infty)$</div>


title: the-function-fxsin-xk-xc-where-k-and-c-are-constants-decreases-always-when
slug: the-function-fxsin-xk-xc-where-k-and-c-are-constants-decreases-always-when-71713
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P2.db
question:
<div class="question">The function \(f(x)=\sin x-k x-c\), where \(k\) and \(c\) are constants, decreases always when</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(k &gt; 1\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(k \geq 1\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(k  &lt; 1\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(k \leq 1\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(k \geq 1\)</span> </div>

answer:
<div class="solution">Let \(f(x)=\sin x-k x-c\) where \(k\) and \(c\) are constants. \(f^{\prime}(x)=\cos x-k\)<br/>\(\therefore f\) decreases if \(\cos x \leq k\)<br/>Thus, \(f(x)=\sin x-k x-c\) decrease always when \(k \geq 1\).</div>


title: the-function-mathrmfmathrmx2-mathrmx33-mathrmx212-mathrmx4-has
slug: the-function-mathrmfmathrmx2-mathrmx33-mathrmx212-mathrmx4-has-91277
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P2.db
question:
<div class="question">The function \(\mathrm{f}(\mathrm{x})=2 \mathrm{x}^3-3 \mathrm{x}^2-12 \mathrm{x}+4\), has</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">two points of local maximum</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">two points of local minimum</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">one maxima and one minima</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">nomaxima or minima</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">one maxima and one minima</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; f(x)=2 x^3-3 x^2-12 x+4 \\<br/>&amp; \Rightarrow \quad f^{\prime}(x)=6 x^2-6 x-12=6\left(x^2-x-2\right) \\<br/>&amp; =6(x-2)(x+1)<br/>\end{aligned}\)<br/>For maxima and minima \(\mathrm{f}^{\prime}(\mathrm{x})=0\)<br/>\(\therefore \quad 6(\mathrm{x}-2)(\mathrm{x}+1)=0 \Rightarrow \mathrm{x}=2,-1\)<br/>Now, \(\mathrm{f}^{\prime \prime}(\mathrm{x})=12 \mathrm{x}-6\)<br/>\(\begin{aligned}<br/>&amp; \text { At } \mathrm{x}=2 ; \mathrm{f}^{\prime \prime}(\mathrm{x})=24-6=18 &gt; 0 \\<br/>&amp; \therefore \quad \mathrm{x}=2, \text { local min. point } \\<br/>&amp; \text { At } \mathrm{x}=-1 ; \mathrm{f}^{\prime \prime}(\mathrm{x})=12(-1)-6=-18  &lt; 0 \\<br/>&amp; \therefore \mathrm{x}=-1 \text { local max. point }<br/>\end{aligned}\)</div>


title: the-function-mathrmfmathrmxmathrmx1-sqrtell-n-mathrmx-is-at-mathrmx1
slug: the-function-mathrmfmathrmxmathrmx1-sqrtell-n-mathrmx-is-at-mathrmx1-35837
tags: ['Mathematics', 'Continuity and Differentiability', 'BITSAT', 'BITSAT 2011']
answer: <div class="solution">Continuous as well as differentiable, so \( f^{\prime}(1)=0\)</div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The function \(\mathrm{f}(\mathrm{x})=(\mathrm{x}-1) \sqrt{|\ell n \mathrm{x}|}\) is at \(\mathrm{x}=1\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">discontinuous</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">continuous but not differentiable</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">differentiable with \(f^{\prime}(1)=0\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">differentiable with \(\mathrm{f}^{\prime}(1) \neq 0\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">differentiable with \(f^{\prime}(1)=0\)</span> </div>


title: the-function-mathrmfmathrmxtan-mathrmx4-mathrmx-is-strictly-decreasing-on
slug: the-function-mathrmfmathrmxtan-mathrmx4-mathrmx-is-strictly-decreasing-on-65109
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P2.db
question:
<div class="question">The function \(\mathrm{f}(\mathrm{x})=\tan \mathrm{x}-4 \mathrm{x}\) is strictly decreasing on</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\left(-\frac{\pi}{3}, \frac{\pi}{3}\right)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\left(\frac{\pi}{3}, \frac{\pi}{2}\right)\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\left(-\frac{\pi}{3}, \frac{\pi}{2}\right)\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\left(\frac{\pi}{2}, \pi\right)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\left(-\frac{\pi}{3}, \frac{\pi}{3}\right)\)</span> </div>

answer:
<div class="solution">\(f(x)=\tan x-4 x \Rightarrow f^{\prime}(x)=\sec ^2 x-4\)<br/>When \(\frac{-\pi}{3}  &lt; \mathrm{x} &lt; \frac{\pi}{3}, 1 &lt; \sec \mathrm{x} &lt; 2\)<br/>Therefore, \(1 &lt; \sec ^2 x &lt; 4\)<br/>\(\Rightarrow-3 &lt; \left(\sec ^2 x-4\right) &lt; 0\)<br/>Thus, for \(\frac{-\pi}{3} &lt; \mathrm{x} &lt; \frac{\pi}{3}, \mathrm{f}^{\prime}(\mathrm{x}) &lt; 0\)<br/>Hence, \(\mathrm{f}\) is strictly decreasing on \(\left(\frac{-\pi}{3}, \frac{\pi}{3}\right)\)</div>


tags: ['Physics', 'Waves and Sound', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P2.db
title:
the-fundamental-frequency-of-an-open-organ-pipe-is-300-hz-the-first-overtone-of-this-pipe-has-same-frequency-as-first-overtone-of-a-closed-organ-pipe-

slug:
the-fundamental-frequency-of-an-open-organ-pipe-is-300-hz-the-first-overtone-of-this-pipe-has-same-frequency-as-first-overtone-of-a-closed-organ-pipe-36050

question:
<div class="question">The fundamental frequency of an open organ pipe is $300 \mathrm{~Hz}$. The first overtone of this pipe has same frequency as first overtone of a closed organ pipe. If speed of sound is $330 \mathrm{~m} / \mathrm{s}$, then the length of closed organ pipe is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$41 \mathrm{~cm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$37 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$31 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$80 \mathrm{~cm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$41 \mathrm{~cm}$</span> </div>

answer:
<div class="solution">For open pipe, $\mathrm{n}=\frac{\mathrm{v}}{2 \ell},$ where $\mathrm{n}_{0}$ is the<br/>fundamental frequency of open pipe. $\therefore \ell=\frac{\mathrm{v}}{2 \mathrm{n}}=\frac{330}{2 \times 300}=\frac{11}{20}$<br/>As freq. of 1 st overtone of open pipe $=$ freq. of 1 st overtone of closed pipe<br/>$\begin{aligned} \therefore \quad 2 \frac{\mathrm{v}}{2 \ell}=3 \frac{\mathrm{v}}{4 \ell^{\prime}} \Rightarrow \ell^{\prime} &amp;=\frac{3 \ell}{4}=\frac{3}{4} \times \frac{11}{20} \\ &amp;=41.25 \mathrm{~cm} \end{aligned}$</div>


title: the-general-solution-of-8-tan-2-fracx21sec-x-is
slug: the-general-solution-of-8-tan-2-fracx21sec-x-is-22945
question: <div class="question">The general solution of \(8 \tan ^2 \frac{x}{2}=1+\sec x\) is</div>
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(2 n \pi \pm \cos ^{-1}\left(\frac{-1}{3}\right)\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(2 \mathrm{n} \pi \pm \frac{\pi}{6}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(2 \mathrm{n} \pi \pm \cos ^{-1}\left(\frac{1}{3}\right)\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(2 \mathrm{n} \pi \pm \cos ^{-1}\left(\frac{1}{3}\right)\)</span> </div>

answer:
<div class="solution">We have, \(8 \tan ^2 \frac{x}{2}=1+\sec x\)<br/>\(\begin{aligned}<br/>&amp; \Rightarrow 8\left(\frac{1-\cos x}{1+\cos x}\right)=1+\frac{1}{\cos x}=\frac{1+\cos x}{\cos x} \\<br/>&amp; \Rightarrow 8 \cos x-8 \cos ^2 x=(1+\cos x)^2 \\<br/>&amp; \Rightarrow 9 \cos ^2 \mathrm{x}-6 \cos \mathrm{x}+1=0 \\<br/>&amp; \Rightarrow(3 \cos \mathrm{x}-1)^2=0 \Rightarrow 3 \cos \mathrm{x}-1=0 \\<br/>&amp; \Rightarrow \cos \mathrm{x}=\frac{1}{3}=\cos \alpha \text { (say) }  \Rightarrow \mathrm{x}=2 \mathrm{n} \pi \pm \alpha \\<br/>&amp; \therefore \mathrm{x}=2 \mathrm{n} \pi \pm \cos ^{-1}\left(\frac{1}{3}\right) \text { where } \mathrm{n} \in \mathrm{I}<br/>\end{aligned}\)</div>


title: the-general-solution-of-sin-x-3-sin-2-x-sin-3-x-cos-x-3-cos-2-x-cos-3-x-is
slug: the-general-solution-of-sin-x-3-sin-2-x-sin-3-x-cos-x-3-cos-2-x-cos-3-x-is-75830
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P2.db
question:
<div class="question">The general solution of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mi>x</mi><mo>-</mo><mn>3</mn><mi>sin</mi><mn>2</mn><mi>x</mi><mo>+</mo><mi>sin</mi><mn>3</mn><mi>x</mi><mo>=</mo><mi>cos</mi><mi>x</mi><mo>-</mo><mn>3</mn><mi>cos</mi><mn>2</mn><mi>x</mi><mo>+</mo><mi>cos</mi><mn>3</mn><mi>x</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>π</mi><mo>+</mo><mfrac><mi>π</mi><mn>8</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>n</mi><mi>π</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>π</mi><mn>8</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mi>n</mi></msup><mfrac><mrow><mi>n</mi><mi>π</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>π</mi><mn>8</mn></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>h</mi><mi>π</mi><mo>+</mo><msup><mi>cos</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mn>3</mn><mn>2</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>n</mi><mi>π</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>π</mi><mn>8</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mi>x</mi><mo>-</mo><mn>3</mn><mi>sin</mi><mn>2</mn><mi>x</mi><mo>+</mo><mi>sin</mi><mn>3</mn><mi>x</mi><mo>=</mo><mi>cos</mi><mi>x</mi><mo>-</mo><mn>3</mn><mi>cos</mi><mn>2</mn><mi>x</mi><mo>+</mo><mi>cos</mi><mn>3</mn><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>sin</mi><mi>x</mi><mo>+</mo><mi>sin</mi><mn>3</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>sin</mi><mn>2</mn><mi>x</mi><mo>=</mo><mi>cos</mi><mi>x</mi><mo>+</mo><mi>cos</mi><mn>3</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>cos</mi><mn>2</mn><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mi>sin</mi><mn>2</mn><mi>x</mi><mi>cos</mi><mi>x</mi><mo>-</mo><mn>3</mn><mi>sin</mi><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>cos</mi><mn>2</mn><mi>x</mi><mi>cos</mi><mi>x</mi><mo>+</mo><mn>3</mn><mi>cos</mi><mn>2</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>sin</mi><mn>2</mn><mi>x</mi><mo>(</mo><mn>2</mn><mi>cos</mi><mi>x</mi><mo>-</mo><mn>3</mn><mo>)</mo><mo>-</mo><mi>cos</mi><mn>2</mn><mi>x</mi><mo>(</mo><mn>2</mn><mi>cos</mi><mi>x</mi><mo>-</mo><mn>3</mn><mo>)</mo><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>(</mo><mi>sin</mi><mn>2</mn><mi>x</mi><mo>-</mo><mi>cos</mi><mn>2</mn><mi>x</mi><mo>)</mo><mo>(</mo><mn>2</mn><mi>cos</mi><mi>x</mi><mo>-</mo><mn>3</mn><mo>)</mo><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>sin</mi><mn>2</mn><mi>x</mi><mo>=</mo><mi>cos</mi><mn>2</mn><mi>x</mi><mo>  </mo><mo>[</mo><mo>∵</mo><mi>cos</mi><mi>x</mi><mo>≠</mo><mn>3</mn><mo>/</mo><mn>2</mn><mo>]</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>2</mn><mi>n</mi><mi>π</mi><mo>±</mo><mfenced><mrow><mfrac><mi>π</mi><mn>2</mn></mfrac><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mrow><mi>n</mi><mi>π</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>π</mi><mn>8</mn></mfrac></math> [<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo></math> neglect <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>ve</mi></math> sign]</p></div>


title: the-general-solution-of-the-differential-equation-d-x-d-y-sin-x-y-sin-x-y-is
slug: the-general-solution-of-the-differential-equation-d-x-d-y-sin-x-y-sin-x-y-is-63599
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P2.db
question:
<div class="question">The general solution of the differential equation $\frac{d y}{d x}+\sin (x+y)=\sin (x-y)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\log \tan y+\sin x=\mathrm{C}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\quad \log \tan \frac{y}{2}+2 \sin x=\mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\tan \frac{y}{2}+\log \sin x=\mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad \log \tan \frac{y}{2}+2 \sin x=\mathrm{C}$</span> </div>

answer:
<div class="solution">The equation is,
<br/> <br/>$\frac{d y}{d x}=\sin (x-y)-\sin (x+y)=2 \cos x \sin (-y)$
<br/> <br/>$\Rightarrow \frac{d y}{\sin y}+2 \cos x d x=0$
<br/> <br/>$\Rightarrow \int \cos e c y d y+2 \int \cos x d x=\mathrm{C}$
<br/> <br/>$\Rightarrow \log \tan \frac{y}{2}+2 \sin x=\mathrm{C}$</div>


title: the-general-solution-of-the-differential-equation-tan-1-y-x-d-y-1-y-2-d-x-is
slug: the-general-solution-of-the-differential-equation-tan-1-y-x-d-y-1-y-2-d-x-is-93888
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
question:
<div class="question">The general solution of the differential equation $\left(\tan ^{-1} y-x\right) d y=\left(1+y^{2}\right) d x$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x=\left(\tan ^{-1} y+1\right)+\mathrm{C} e^{-\tan ^{-1} y}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$x=\left(\tan ^{-1} y-1\right)+\mathrm{C} e^{-\tan ^{-1} y}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x=\left(\tan ^{-1} x-1\right)+\mathrm{C} e^{-\tan ^{-1} x}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x=\left(\tan ^{-1} x+1\right)+\mathrm{Ce}^{-\tan ^{-1} x}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x=\left(\tan ^{-1} y-1\right)+\mathrm{C} e^{-\tan ^{-1} y}$</span> </div>

answer:
<div class="solution">The given differential equation can be written as $\frac{d x}{d y}+\frac{x}{1+y^{2}}=\frac{\tan ^{-1} y}{1+y^{2}} \ldots$ (i)
<br/> <br/>Now, eq. (i) is a linear differential equation of the form $\frac{d x}{d y}+\mathrm{P}_{1} x=\mathrm{Q}_{1}$
<br/> <br/>where $\mathrm{P}_{1}=\frac{1}{1+y^{2}}$ and $\mathrm{Q}_{1}=\frac{\tan ^{-1} y}{1+y^{2}}$
<br/> <br/>Therefore, I.F $=e^{\int \frac{1}{1+y^{2}} d y}=e^{\tan ^{-1} y}$
<br/> <br/>Thus, the solution of the given differential equation is given by $x e^{\tan ^{-1} y}=\int\left(\frac{\tan ^{-1} y}{1+y^{2}}\right) e^{\tan ^{-1} y} \mathrm{dy}+\mathrm{C} \ldots$ (ii)
<br/> <br/>Let $\mathrm{I}=\int\left(\frac{\tan ^{-1} y}{1+y^{2}}\right) e^{\tan ^{-1} y} \mathrm{dy}$
<br/> <br/>On substituting $\tan ^{-1} y=t$, so that
<br/> <br/>$\left(\frac{1}{1+y^{2}}\right) \mathrm{dy}=\mathrm{dt}$, we get
<br/> <br/>$I=\int t e^{t} d t=t e^{t}-\int 1 \cdot e^{t} d t=t e^{t}-e^{t}=e^{t}(t-1)$
<br/> <br/>or $\mathrm{I}=e^{\tan ^{-1} y}\left(\tan ^{-1} y-1\right)$
<br/> <br/>On substituting the value of 1 in equation (i1), we get
<br/> <br/>$x \cdot e^{\tan ^{-1} y}=\cdot e^{\tan ^{-1} y}\left(\tan ^{-1} y-1\right)+\mathrm{C}$
<br/> <br/>or $x=\left(\tan ^{-1} y-1\right)+\mathrm{C} e^{\tan ^{-1} y}$
<br/> <br/>which is the general solution of the given differntial equation.</div>


title: the-general-solution-of-the-equation-sin-2-x-2-sin-x-2-cos-x-1-0-is
slug: the-general-solution-of-the-equation-sin-2-x-2-sin-x-2-cos-x-1-0-is-18437
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
question:
<div class="question">The general solution of the equation $\sin 2 x+2 \sin x+2 \cos x+1=0$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 \mathrm{n} \pi-\frac{\pi}{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \mathrm{n} \pi+\frac{\pi}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \mathrm{n} \pi+(-1)^{\mathrm{n}} \sin ^{-1}\left(\frac{1}{\sqrt{3}}\right)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{n} \pi-\frac{\pi}{4}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{n} \pi-\frac{\pi}{4}$</span> </div>

answer:
<div class="solution">Given, $\sin 2 \mathrm{x}+2 \sin \mathrm{x}+2 \cos \mathrm{x}+1=0$
<br/> <br/>$\Rightarrow 1+\sin 2 \mathrm{x}+2(\sin \mathrm{x}+\cos \mathrm{x})=0$
<br/> <br/>$\Rightarrow(\sin x+\cos x)^{2}+2(\sin x+\cos x)=0$
<br/> <br/>$\Rightarrow(\sin x+\cos x)(\sin x+\cos x+2)=0$
<br/> <br/>$\therefore \sin x+\cos x=0$ or $\sin x+\cos x=-2$
<br/> <br/>But, $\sin x+\cos x=-2$ is inadmissible.
<br/> <br/>Since, $|\sin x| \leq 1,|\cos x| \leq 1$
<br/> <br/>$\therefore \sin x+\cos x=0 \Rightarrow \sin \left(x+\frac{\pi}{4}\right)=0$
<br/> <br/>$\Rightarrow x+\frac{\pi}{4}=n \pi \Rightarrow x=n \pi-\frac{\pi}{4}$</div>


tags: ['Chemistry', 'States of Matter', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P2.db
title:
the-given-graph-represents-the-variation-of-compressibility-factor-z-n-rt-p-v-for-three-real-gases-a-b-and-c-identify-the-only-incorrect-statement

slug:
the-given-graph-represents-the-variation-of-compressibility-factor-z-n-rt-p-v-for-three-real-gases-a-b-and-c-identify-the-only-incorrect-statement-62307

question:
<div class="question">The given graph represents the variation of compressibility factor $(Z)=\frac{p V}{n R T}$, for three real gases $A, B$ and $C$. Identify the only incorrect statement.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/oyRA5keONN_o_pg7YjXunRyP45aYxtUbrNo4PgoPv30.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>For the gas $A, a=0$ and its dependence on $p$ is linear at all pressure.
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>For the gas $B, b=0$ and its dependence on $p$ is linear at all pressure.
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>For the gas $C$, which is typical real gas for which neither. $a$ nor $b=0$. By knowing the minima and point of the intersection, with $Z=1, a$ and $b$ can be calculated.
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>At high pressure the slope is positive for all real gases.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>For the gas $B, b=0$ and its dependence on $p$ is linear at all pressure.
<br/></span> </div>

answer:
<div class="solution">From the graph it is clear that, the value of ' $Z$ ' decreases with increase of pressure. We can explain as follows on the basis of van der Waals' equation.
<br/>At high pressure, when ' $p$ ' is large, $V$ will be small and one cannot ignore ' $b$ ' in comparison to $V$. However, the team $a / V^2$ may be considered negligible in comparison to ' $p$ ' in van der Waals' equation.
<br/>$\left(p+\frac{a}{V^2}\right)(V-b)=n R T$
<br/>$p(V-b)=n R T$
<br/>$\Rightarrow \quad p V-p b=n R T$
<br/>or    $\frac{p V}{n R T}=1+\frac{p b}{n R T}$
<br/>or $Z=1+\frac{p b}{n R T}$
<br/>Thus, $Z$ is greater than 1. As pressure is increased (at constant $T$ ), the factor $\frac{p b}{n R T}$ increases. This explains why after minima in the curves, $Z$ increase continuously with pressure. Hence, the only incorrect statement is (b).</div>


tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P2.db
title:
the-given-lens-is-broken-into-four-parts-and-rearranged-as-shown-if-the-initial-focal-length-is-f-then-after-rearrangement-the-equivalent-focal-length

slug:
the-given-lens-is-broken-into-four-parts-and-rearranged-as-shown-if-the-initial-focal-length-is-f-then-after-rearrangement-the-equivalent-focal-length-66497

question:
<div class="question">The given lens is broken into four parts and rearranged as shown. If the initial focal length is $f$ then after rearrangement the equivalent focal length is-
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/iaDcJLPT7cdlw7MhByr5aEMjKOw_fw4MqXt7of4IIWM.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$f$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$f / 2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$f / 4$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 f$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f / 2$</span> </div>

answer:
<div class="solution">Cutting a lens in transverse direction doubles their focal length i.e. $2 f$. Using the formula of equivalent focal length, $\frac{1}{f}=\frac{1}{f_{1}}+\frac{1}{f_{2}}+\frac{1}{f_{3}}+\frac{1}{f_{4}}$
<br/> <br/>We get equivalent focal length as $f / 2$</div>


